MIMO Communication for Cellular Networks

Howard Huang Constantinos B. PapadiasSivarama Venkatesan

MIMO Communicationfor Cellular Networks

ISBN 978-0-387-77521-0 e-I SBN 978-0-387-77523-4DOI 10.1007/978- - - - Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942318 © Springer Science+Business Media, LLC 2012

Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

0 387 77523 4

Constantinos B. Papadias Athens Information Technology (AIT) Markopoulo Avenue 19.5 km 190 02 Peania, Athens Greece [email protected]

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computerware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

soft-

Howard Huang Bell Labs, Alcatel-Lucent 791 Holmdel Road Holmdel, New Jersey 07733 USA [email protected] Sivarama Venkatesan Bell Labs, Alcatel-Lucent 791 Holmdel Road Holmdel, New Jersey 07733 USA [email protected]

For Michelle, H.H.

For Maria-Anna, C.B.P.

For my parents, S.V.

Preface

Since the 1980s, commercial cellular networks have evolved over several gen-

erations, from providing simple voice telephony service to supporting a wide

range of other applications (such as text messaging, web browsing, streaming

media, social networking, video calling, and machine-to-machine communi-

cation) that are accessed through a variety of devices (such as smart phones,

laptops, tablet devices, and wireless sensors). These developments have fueled

a demand for higher spectral efficiency so that the limited spectral resources

allocated for cellular networks can be utilized more effectively.

In parallel, starting in the mid-1990s [1], multiple-input multiple-output

(MIMO) wireless communication has emerged as one of the most fertile ar-

eas of research in information and communication theory. The fundamental

results of this research show that MIMO techniques have enormous potential

to improve the spectral efficiency of wireless links and systems. These tech-

niques have already attracted considerable attention in the cellular world,

where simple MIMO techniques are already appearing in commercial prod-

ucts and standards, and more sophisticated ones are actively being pursued.

Goals of the book

In this book, we hope to connect these two worlds of MIMO communication

theory and cellular network design with the goal of understanding how multi-

ple antennas can best be used to improve the physical-layer performance of a

cellular system. We attempt to strike a balance between fundamental theoret-

ical results, practical techniques and core insights regarding the performance

limits of multiple antennas in multiuser networks. Unlike books that focus on

the theoretical performance of abstract MIMO channels, this one emphasizes

the practical performance of realistic MIMO systems.

vii

viii Preface

We present in the first part of the book a systematic description of MIMO

capacity and capacity-achieving techniques for different classes of multiple-

antenna channels. The second part of the book describes a framework for

MIMO system design that accounts for the essential physical-layer features

of practical cellular networks. By applying the information-theoretic capacity

results to this framework, we present a unified set of system simulation stud-

ies that highlight relative performance gains of different MIMO techniques

and provides insights into how best to utilize multiple antennas in cellular

networks under various conditions. Characterizations of the system-level per-

formance are provided with sufficient generality that the underlying concepts

can be applied to a wide range of wireless systems, including those based on

cellular standards such as LTE, LTE-Advanced, WiMAX, and WiMAX2.

Intended audience

The book is intended for graduate students, researchers, and practicing engi-

neers interested in the physical-layer design of contemporary wireless systems.

The material is presented assuming the reader is comfortable with linear alge-

bra, probability theory, random processes, and basic digital communication

theory. Familiarity with wireless communication and information theory is

helpful but not required.

Acknowledgements

We have attempted to represent in this book a small sliver of knowledge

accumulated by a vast community of researchers. Over the years, we have

had the great pleasure of learning from and interacting with many mem-

bers of this community within Bell Labs, Alcatel-Lucent, and at other cor-

porations and academic institutions. These experts include Angela Alex-

iou, Alexei Ashikhmin, Dan Avidor, Matthew Baker, Krishna Balachan-

dran, Liyu Cai, Len Cimini, David Goodman, Inaki Esnaola,Vinko Erceg,

Rodolfo Feick, Jerry Foschini, Mike Gans, David Gesbert, Maxime Guillaud,

Bert Hochwald, Syed Jafar, Nihar Jindal, Volker Jungnickel, Kemal Karakay-

ali, Achilles Kogiantis, Alex Kuzminskiy, Persa Kyritsi, Angel Lozano, Mike

MacDonald, Narayan Mandayam, Laurence Mailaender, Thomas Marzetta,

Thomas Michel, Pantelis Monogioudis, Francis Mullany, Marty Meyers, Ar-

ogyaswami Paulraj, Farrokh Rashid-Farrokhi, Niranjay Ravindran, Gee Rit-

tenhouse, Dragan Samardzija, James Seymour, Sana Sfar, Steve Simon, Tod

Sizer, John Smee, Max Solondz, Robert Soni, Aleksandr Stoylar, Said Tatesh,

Stephan ten Brink, Lars Thiele, Filippo Tosato, Cuong Tran, Matteo Trivel-

Preface ix

lato, Giovanni Vannucci, Sergio Verdu, Harish Viswanathan, Sue Walker, Carl

Weaver, Thorsten Wild, Stephen Wilkus, Peter Wolniansky, Greg Wright,

Gerhard Wunder, Hao Xu, Hongwei Yang, Roy Yates, and Mike Zierdt.

Special thanks go to Antonia Tulino who graciously illuminated various

aspects of information theory at a moment’s notice. We would also like to

specifically acknowledge the following colleagues who provided valuable feed-

back on an earlier draft: George Alexandropoulos, Federico Boccardi, Dmitry

Chizhik, Jonathan Ling, Mohammad Ali Maddah-Ali, Chris Ng, Stelios Pa-

paharalabos, Xiaohu Shang, and Marcos Tavares. We would like to express

our gratitude to Francesca Simkin for her keen eye and expert skill in copy

editing the manuscript and to Kimberly Howie for providing tips on improv-

ing the visual design. Allison Michael and Alex Greene at Springer provided

valuable assistance in the production of the manuscript. Finally, we would

like to acknowledge our colleague Reinaldo Valenzuela whose enthusiastic

leadership has helped shape the spirit and goals of this book.

Howard Huang and Sivarama Venkatesan

Bell Labs, Alcatel-Lucent

Holmdel, New Jersey

Constantinos B. Papadias

Athens Information Technology

Athens, Greece

August 2011

Notation

COL SU-MIMO open-loop capacity

COL SU-MIMO average open-loop capacity

CCL SU-MIMO closed-loop capacity

CCL SU-MIMO average closed-loop capacity

CMAC Multiple-access channel capacity region

CMAC Multiple-access channel sum-capacity

CMAC Multiple-access channel average sum-capacity

CBC Broadcast channel capacity region

CBC Broadcast channel sum-capacity

CBC Broadcast channel average sum capacity

CTDMA TDMA channel maximum achievable sum rate

CTDMA TDMA channel average maximum achievable sum rate

M SU-MIMO channel: number of transmit antennas

MU-MIMO MAC: number of receive antennas

MU-MIMO BC: number of transmit antennas

Cellular system: number of antennas per base

N SU-MIMO channel: number of receive antennas

MU-MIMO MAC: number of transmit antennas per user

MU-MIMO BC: number of receive antennas per user

Cellular system: number of antennas per user

K Number of users per base

B Number of bases

S Number of sectors per site

H,h Complex-valued channel matrix, vector

s Transmitted signal vector

Q Transmitted signal covariance

x Received signal vector

G,g Precoding matrix, vector

xi

xii Notation

n Received noise vector

u Vector of data symbols

P Signal power constraint

σ2 Noise variance

P/σ2 SU- and MU-MIMO channel: average SNR

Cellular system: reference SNR

λ2max(H) Maximum eigenvalue of HHH

v Symbol power

q Quality-of-service weight

α2 Average channel gain

d Distance

dref Reference distance

Z Shadow fading realization

G Directional antenna response

γ Pathloss coefficient

Γ Geometry

E(x) Expected value of random variable x

AH Hermitian transpose of matrix A

trA Trace of square matrix A

diagA Diagonal elements of square matrix A

diag(a1, . . . , aN ) Square N ×N matrix with diagonal elements a1, . . . , aN

IN N ×N identity matrix

0N N × 1 vector of zeroes

C Set of complex numbers

R Set of real numbers

B Set of precoding matrices

U Set of active users

Contents

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview of MIMO fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 MIMO channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Single-user capacity metrics . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Multiuser capacity metrics . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.4 MIMO performance gains . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Overview of cellular networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 System characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.2 Co-channel interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3 Overview of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Single-user MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1 Channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Analytical channel models . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.2 Physical channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.3 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Single-user MIMO capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 Capacity for fixed channels . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.2 Performance gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2.3 Performance comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3 Transceiver techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1 Linear receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2 MMSE-SIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3.3 V-BLAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xiii

viiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .

xiv Contents

2.3.4 D-BLAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.3.5 Closed-loop MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3.6 Space-time coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3.7 Codebook precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.4 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.4.1 CSI estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.4.2 Spatial richness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 Multiuser MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1 Channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.2 Multiple-access channel (MAC) capacity region . . . . . . . . . . . . . 81

3.2.1 Single-antenna transmitters (N = 1) . . . . . . . . . . . . . . . . 82

3.2.2 Multiple-antenna transmitters (N > 1) . . . . . . . . . . . . . . 85

3.3 Broadcast channel (BC) capacity region . . . . . . . . . . . . . . . . . . . 87

3.3.1 Single-antenna transmitters (M = 1) . . . . . . . . . . . . . . . . 89

3.3.2 Multiple-antenna transmitters (M > 1) . . . . . . . . . . . . . 91

3.4 MAC-BC Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.5 Scalar performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5.1 Maximizing the MAC weighted sum rate . . . . . . . . . . . . 96

3.5.2 Maximizing the BC weighted sum rate . . . . . . . . . . . . . . 99

3.6 Sum-rate performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.6.1 MU-MIMO sum-rate capacity and SU-MIMO capacity 103

3.6.2 Sum rate versus SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.6.3 Sum rate versus the number of base antennas M . . . . . 109

3.6.4 Sum rate versus the number of users K . . . . . . . . . . . . . 113


3.7.1 Successive interference cancellation . . . . . . . . . . . . . . . . . 117

3.7.2 Dirty paper coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.7.3 Spatial richness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4 Suboptimal Multiuser MIMO Techniques . . . . . . . . . . . . . . . . . 121

4.1 Suboptimal techniques for the multiple-access channel . . . . . . . 121

4.1.1 Beamforming for the case of many users . . . . . . . . . . . . . 122

4.1.2 Alternatives for MMSE-SIC detection . . . . . . . . . . . . . . . 124

4.2 Suboptimal techniques for the broadcast channel . . . . . . . . . . . 125

4.2.1 CSIT precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Contents xv

4.2.2 CDIT precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.2.3 Codebook precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.2.4 Random precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.3 MAC-BC duality for linear transceivers . . . . . . . . . . . . . . . . . . . 149

4.3.1 MAC power control problem . . . . . . . . . . . . . . . . . . . . . . . 151

4.3.2 BC power control problem . . . . . . . . . . . . . . . . . . . . . . . . . 152


4.4.1 Obtaining CSIT in FDD and TDD systems . . . . . . . . . . 153

4.4.2 Reference signals for channel estimation . . . . . . . . . . . . . 154

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5 Cellular Networks: System Model and Methodology . . . . . . 159

5.1 Cellular network design components . . . . . . . . . . . . . . . . . . . . . . 160

5.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.2.1 Sectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.2.2 Reference SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2.3 Cell wraparound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.3 Modes of base station operation . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.3.1 Independent bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.3.2 Coordinated bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.4 Simulation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.4.1 Performance criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.4.2 Iterative scheduling algorithm . . . . . . . . . . . . . . . . . . . . . 190

5.4.3 Methodology for proportional-fair criterion . . . . . . . . . . 194

5.4.4 Methodology for equal-rate criterion . . . . . . . . . . . . . . . . 202


5.5.1 Antenna array architectures . . . . . . . . . . . . . . . . . . . . . . . 213

5.5.2 Sectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.5.3 Signaling to support MIMO . . . . . . . . . . . . . . . . . . . . . . . 218

5.5.4 Scheduling for packet-switched networks . . . . . . . . . . . . . 219

5.5.5 Acquiring CQI, CSIT, and PMI . . . . . . . . . . . . . . . . . . . . 222

5.5.6 Coordinated base stations . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

6 Cellular Networks: Performance and Design Strategies . . . 227

6.1 Simulation assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.2 Isolated cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.3 Cellular network with independent single-antenna bases . . . . . 232

xvi Contents

6.3.1 Universal frequency reuse, single sector per site . . . . . . 233

6.3.2 Throughput per area versus cell size . . . . . . . . . . . . . . . . 236

6.3.3 Reduced frequency reuse performance . . . . . . . . . . . . . . . 238

6.3.4 Sectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

6.4 Cellular network with independent multiple-antenna bases . . . 249

6.4.1 Throughput scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6.4.2 Fixed number of antennas per site . . . . . . . . . . . . . . . . . . 251

6.4.3 Directional beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.4.4 Increasing range with MIMO . . . . . . . . . . . . . . . . . . . . . . 267

6.5 Cellular network, coordinated bases . . . . . . . . . . . . . . . . . . . . . . . 269

6.5.1 Coordinated precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.5.2 Network MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272


6.7 Cellular network design strategies . . . . . . . . . . . . . . . . . . . . . . . . 280

7 MIMO in Cellular Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7.1 UMTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

7.2 LTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

7.3 LTE-Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.4 IEEE 802.16e and IEEE 802.16m . . . . . . . . . . . . . . . . . . . . . . . . . 295

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 309

Chapter 1

Introduction

In a cellular network, a given geographic area is served by several base sta-

tions, as shown in Figure 1.1. Each base station (or, simply, base) commu-

nicates wirelessly using radio-frequency spectrum with one or more mobile

terminals (or users) assigned to it. Conceptually, it is useful to think of the

geographic area as being partitioned into cells, with each cell representing

the area served by one of the bases. Two-way wireless communication occurs

between the users and bases. On the uplink, a base detects signals from its

assigned users; on the downlink, a base transmits signals to its assigned users.

A system engineer’s job is to design the cellular network to provide reli-

able wireless communication using the base station assets over the allocated

spectrum resources, where the transmissions are subject to constraints on the

radiated power. The network performance measure of most interest to us is

the throughput, defined as the total data rate transmitted or received by a

base station, measured in bits per second (bps). Since we are interested in

the performance for a given channel bandwidth, we will focus in this book

on the throughput spectral efficiency (measured in bps per Hertz) and the

spectral efficiency per unit area (measured in bps per Hertz per square kilo-

meter). The emphasis on spectral efficiency is justified by the scarcity and

consequent high cost of radio spectrum. Improvements in spectral efficiency

can translate into other metrics including higher data rates to the user, im-

proved coverage, improved service reliability, and lower network cost to the

operator.

The goal of this book is to explore how the physical-layer performance of

contemporary cellular networks can be improved using multiple antennas at

the bases and user terminals. Multiple antenna techniques, or multiple-input

multiple-output (MIMO) techniques, allow us to exploit the spatial dimen-

1H. Huang et al., MIMO Communication for Cellular Networks, DOI 10.1007/978-0-387-77523-4_1, © Springer Science+Business Media, LLC 2012

2 1 Introduction

wefawef

Mobile terminal (user)

Base station (base)

Uplink transmission

Downlink transmission

Fig. 1.1 In a cellular network, the geographic area is partitioned into cells, and base

stations communicate wirelessly with their assigned users.

sion of the wireless channel, resulting in benefits that include robustness

against channel fading, gains in the desired signal power, protection against

co-channel interference, and reuse of the spectral resources. MIMO techniques

affect the spectral efficiency to varying degrees, and we will investigate the

tradeoffs from a system-level perspective between performance gains and im-

plementation complexity for the different techniques.

To begin, we introduce in Section 1.1 some fundamentals of MIMO tech-

niques in the context of isolated channels. In Section 1.2 we give an overview

of cellular networks and describe how multiple antennas could be used in this

system-level context.

1.1 Overview of MIMO fundamentals

The wireless links in a cellular network can be characterized as either single-

user channels or multiuser channels, as shown in Figure 1.2. In this section, we

describe some simple models for these channels and give an overview of their

theoretical performance limits. These results are described in more detail in

Chapters 2 and 3.

1.1.1 MIMO channel models

Single-user MIMO channel

The single-user channel models a point-to-point link between a base and a

1.1 Overview of MIMO fundamentals 3

SU-MIMOchannel

Broadcast channel (BC)

Multiple accesschannel (MAC)

...

...

Single-user (SU) MIMO Multiuser (MU) MIMO

Fig. 1.2 Single-user MIMO communication occurs over a point-to-point link. Multiuser

MIMO communication occurs over a multipoint-to-point link (multiple-access channel) or

a point-to-multipoint link (broadcast channel).

user. For the downlink, the base is the transmitter and the user is the receiver.

For the uplink, the roles are reversed.

We define an (M,N) single-user (SU) MIMO channel as a communication

link with M ≥ 1 antennas at the transmitter and N ≥ 1 antennas at the

receiver. (Special cases of the (M,N) MIMO channel are the (M, 1) multiple-

input, single-output (MISO) channel, the (1, N) single-input, multiple-output

(SIMO) channel, and the (1,1) single-input, single-output (SISO) channel.)

The baseband received signal at a given antenna is a linear combination of the

M transmitted signals, each modulated by the channel’s complex amplitude

coefficient, and corrupted by noise. The baseband signal received over a SU-

MIMO channel for the duration of a symbol period can be written using

vector notation:

x = Hs+ n, (1.1)

where

• x ∈ CN×1 is the received signal whose nth element (n = 1, . . . , N) is

associated with antenna n

• H ∈ CN×M is the channel matrix whose (n,m)th entry (n = 1, . . . , N ;m =

1, . . . ,M) gives the complex amplitude between the mth transmit antenna

and the nth receive antenna

4 1 Introduction

• s ∈ CM×1 is the transmitted signal vector, having covariance Q :=

E(ssH), and subject to the power constraint trQ ≤ P

• n ∈ CN×1 is a circularly symmetric complex Gaussian vector representing

additive receiver noise, with mean E(n) = 0N and covariance E(nnH) =

σ2IN

In practice, the wireless channel experiences fading in both the frequency

and time domains. However, for simplicity we assume that the channel band-

width is small compared to the coherence bandwidth so that the channel is

frequency-nonselective. In environments where the transmitter and receiver

are stationary, we can model H as being fixed for the duration of a coding

block (on the order of hundreds or thousands of symbol periods) but changing

randomly from one block to another. If the antennas at the transmitter and

receiver are spaced sufficiently far apart (with respect to the channel angle

spread), the elements of H can be modeled as realizations of independent

and identically distributed (i.i.d.) Gaussian random variables with zero mean

and unit variance. Because the amplitude of each element has a Rayleigh

distribution, this channel is said to be drawn from an i.i.d. Rayleigh distri-

bution. Because the channel coefficients of H are independent and therefore

uncorrelated, the channel is said to be spatially rich. If the elements of H

have unit variance, then the average signal-to-noise ratio (SNR) at any of the

N receive antennas is P/σ2.

Multiuser MIMO channels

We consider two types of multiuser channels: the multiple-access channel

(MAC) and the broadcast channel (BC).

The MAC is used to model a single base receiving signals from multiple

users on the uplink of a cellular network. We will use the notation ((K,N),M)

to denote a MAC with K ≥ 1 users, each with N ≥ 1 antennas, whose signals

are received by a base with M ≥ 1 antennas. We use the term MIMO MAC

to denote a MAC with multiple (M > 1) base antennas serving multiple

(K > 1) users, where each user has one or more antennas.

The data signals sent by the users are independent, and the base receives

the sum of K signals modulated by each user’s MIMO channel and corrupted

by noise. The baseband received signal can be written as:

x =

K∑k=1

Hksk + n, (1.2)


where

• x ∈ CM×1 is the received signal whose mth element (m = 1, . . . ,M) is

associated with antenna m

• Hk ∈ CM×N is the channel matrix for the kth user (k = 1, . . . ,K) whose

(m,n)th entry (m = 1, . . . ,M ;n = 1, . . . , N) gives the complex ampli-

tude between the nth transmit antenna of user k and the mth receive

antenna. Each coefficient for the kth user’s channel Hk is assumed to be

i.i.d. Rayleigh with unit variance, and the channels are mutually indepen-

dent among the users

• sk ∈ CN×1 is the transmitted signal vector from user k. The covariance is

Qk := E(sks

Hk

), and the signal is subject to the power constraint trQk ≤

Pk

• n ∈ CM×1 is a circularly symmetric complex Gaussian vector representing

additive receiver noise, with mean E(n) = 0M and covariance E(nnH) =

σ2IM

The BC is used to model a single base serving multiple users in the down-

link of a cellular network. Each user receives independent data, so the infor-

mation is not broadcast in the sense of users receiving the same data. We use

the shorthand (M, (K,N)) to denote a base station with M ≥ 1 transmit

antennas serving K ≥ 1 users, each with N ≥ 1 antennas. We use the term

MIMO BC to denote a BC with multiple (M > 1) base antennas serving

multiple (K > 1) users, where each user has one or more antennas. The base

transmits a common signal s, which contains the encoded symbols of the K

users’ data streams. This signal travels over the channel HHk to reach user k

(k = 1, . . . ,K). (The Hermitian transpose allows for convenient comparisons

between the MAC and BC, as we will see later in Chapter 3.) The baseband

received signal by the kth user can be written as:

xk = HHk s+ nk, (1.3)

where

• xk ∈ CN×1 whose nth element (n = 1, . . . , N) is associated with antenna

n of user k

• HHk ∈ C

N×M is the channel matrix for the kth user (k = 1, . . . ,K) whose

(n,m)th entry (n = 1, . . . , N ;m = 1, . . . ,M) gives the complex amplitude

between the nth transmit antenna of user k and the mth receive antenna.

Each coefficient for the kth user’s channel HHk is assumed to be i.i.d.

6 1 Introduction

Rayleigh with unit variance, and the channels are mutually independent

among the users

• s ∈ CM×1 is the transmitted signal vector, which is a function of the data

signals for the K users. The covariance is Q := E(ssH), and the signal is

subject to the power constraint trQ ≤ P

• nk ∈ CN×1 is a circularly symmetric complex Gaussian vector representing

additive receiver noise, with mean E(n) = 0N and covariance E(nnH) =

σ2IN

1.1.2 Single-user capacity metrics

For a single-user channel, the performance metric of interest is the spectral

efficiency, defined as the data rate achieved per unit of bandwidth and mea-

sured in units of bits per second per Hertz. When it is understood that the

bandwidth is held fixed, we will often use rate and spectral efficiency in-

terchangeably. From information theory, the Shannon capacity (or simply,

capacity) of the SU-MIMO channel is the maximum spectral efficiency at

which reliable communication is possible, meaning that the bit error rate

can be made arbitrarily small with sufficiently long coding blocks over many

symbol periods. Here we describe the capacity of the SISO, SIMO, MISO,

and MIMO channels when the channels are time-invariant. Rather than use

information theory to derive the capacity, we instead focus on the spectral

efficiency of techniques for approaching the capacity limits in practice.

1.1.2.1 SISO capacity

From (1.1), the received signal over a SISO channel can be written as

x = hs+ n, (1.4)

where h is the fixed scalar complex amplitude of the channel. A sufficient

statistic for detecting s is obtained by multiplying the received signal by the

complex conjugate of h to yield

h∗x = |h|2 s+ h∗n. (1.5)


The sufficient statistic has an effective SNR of |h|2 P/σ2, and the capacity of

this channel is

C = log2

(1 +

P |h|2σ2

)bps/Hz. (1.6)

In Gaussian channels, one can achieve spectral efficiency very close to capacity

using contemporary codes such as turbo codes or low-density parity check

(LDPC) codes in combination with iterative decoding algorithms. We will

loosely refer to such codes as capacity-achieving (although near-capacity-

achieving would be more accurate). Since (1.5) is equivalent to a Gaussian

channel, we can (nearly) achieve capacity using these optimal codes if the

channel h is known ideally at the receiver to provide the sufficient statistic.

1.1.2.2 SIMO capacity

We now consider a (1, N) SIMO channel, with N antennas at the receiver:

x = hs+ n, (1.7)

where h ∈ CN×1 is the time-invariant SIMO channel. A sufficient statistic

is obtained by taking the inner product of the received signal x with the

channel h:

hHx = ‖h‖2 s+ hHn. (1.8)

In doing so, the SIMO channel is reduced to a scalar Gaussian channel. This

linear receiver combining maximizes the output SNR, and it is known as

maximal ratio combining (MRC). The SNR is ‖h‖2 P/σ2, and the capacity

is

C = log2

(1 +

P ‖h‖2σ2

)bps/Hz. (1.9)

The capacity of the (1, N) SIMO channel can be achieved using optimal

capacity-achieving codes and a linear combiner hH as shown in the top half

of Figure 1.3. As a result of combining, the multiple receive antennas provide

a power gain in the output SNR. For example, if the absolute value of each

channel coefficient is 1 (|hn| = 1 for n = 1, . . . , N), then ‖h‖2 = N , and the N

antennas provide a power gain of N compared to the SISO channel. Similarly,

if the channel elements are i.i.d. Rayleigh (with unit mean variance), then

the average power gain is E(‖h‖2) = N .

8 1 Introduction

The impact of the power gain on the capacity (1.9) depends on the channel

SNR P/σ2. We can use the following expressions to approximate the capacity

performance for very high and very low SNRs:

log2(1 + x) ≈ log2(x) if x � 1 (1.10)

log2(1 + x) ≈ x log2(e) if x � 1. (1.11)

For the case where the channel SNR P/σ2 is very high, we see from (1.10) that

doubling the output SNR due to combining results in a 1 bps/Hz increase

in the capacity. In other words, if the output SNR increases linearly, the

capacity increases logarithmically. On the other hand if the channel SNR is

very low, we see from (1.11) that a linear increase in the output SNR results

in a linear increase in the capacity. Therefore multiple antennas in the SIMO

channel boost the capacity more significantly at low SNRs.

M

M

Data bits

Data bits

Coding,modulation

Demod,decoding

Demod,decoding

Coding,modulation

SIMO channel

MISO channel

Linearprecoding

Linearcombining

Fig. 1.3 Capacity is achieved over a SIMO channel with linear combining matched to

the channel h ∈ CN×1. Capacity is achieved over a MISO channel with a linear precoder

matched to the channel h ∈ C1×M .

1.1.2.3 MISO capacity

For an (M, 1) MISO channel (h ∈ C1×M ), where h is known at the transmit-

ter, the capacity is

C = log2

(1 +

P ‖h‖2σ2

)bps/Hz. (1.12)

This capacity can be achieved using the structure shown in bottom half of

Figure 1.3. The transmitted signal at antenna m (m = 1, . . . ,M) is sm =


gmu, where gm is a complex-valued weight applied to an encoded data symbol

u. Weighting the data symbol in this manner at the transmitter is known as

linear precoding, or simply precoding. To achieve the MISO capacity, the

precoding weights are chosen to match the channel h so that the weight on

the mth antennas is the normalized complex conjugate of the mth antenna’s

channel: gm = h∗m/ ‖h‖. If the data symbols have power E(|u|2) = P , then the

transmitted signal also has power P because trE(ssH) = trE(guu∗gH) = P .

The received signal is

x = hs+ n = ‖h‖u+ n, (1.13)

and the SNR is ‖h‖2 P/σ2, yielding the capacity (1.12). This capacity is the

same as the SIMO channel’s (1.9), implying that the power gain achieved

using combining over the SIMO channel can be achieved using precoding

over the MISO channel if the channel is known at the transmitter.

Using precoding with weights matched to the MISO channel is similar

to MRC matched to the SIMO channel. The channel-matched precoding is

sometimes known asmaximal ratio transmission (MRT) because it maximizes

the received SNR. If the channel is line-of-sight (or has a very small angle

spread) and if the transmit antennas are closely spaced (by about half the

carrier wavelength), MRT creates a directional beam pointed towards the

receiver. This type of precoding is sometimes known as transmitter beam-

forming.

1.1.2.4 SU-MIMO capacity

If multiple antennas are used at both the transmitter and receiver, the (M,N)

MIMO channel could support multiple data streams. The data streams are

spatially multiplexed and are sent simultaneously over the same frequency

resources across theM antennas. The receiver usesN antennas to demodulate

the streams based on their spatial characteristics. The capacity of the MIMO

channel depends on knowledge of the channel state information (CSI) H. For

now, we consider the closed-loop MIMO capacity that assume CSI is known

at both the transmitter and receiver. (The open-loop MIMO capacity assumes

that CSI is known only at the receiver and will be discussed in Chapter 2).

If both the transmitter and receiver have ideal knowledge of CSI, the

MIMO channel can be decomposed into r ≥ 1 parallel (non-interfering) scalar

Gaussian subchannels, where r is the rank of the MIMO channel H. For

10 1 Introduction

simplicity, we assume here that H is full rank so that r = min(M,N). This

decomposition is achieved by applying a precoding matrix at the transmitter

and a combining matrix at the receiver. These matrices are derived from

the eigendecomposition of HHH and HHH , respectively, and therefore the

resulting parallel subchannels are often referred to as eigenmodes.

The SNRs of the r scalar Gaussian subchannels turn out to be proportional

to the eigenvalues λ21, λ

22, . . . , λ

2r of HHH (or HHH). More precisely, if the

transmitter allocates power Pj to subchannel j, then the SNR achieved on it

equals λ2jPj/σ

2 and the corresponding data rate is

log2(1 + λ2jPj/σ

2). (1.14)

The capacity of the composite MIMO channel can then be expressed as the

sum of the r subchannel capacities (1.14), maximized over all power alloca-

tions P1, P2, . . . , Pr subject to the constraint∑r

j=1 Pj ≤ P . The resulting

(closed-loop) MIMO capacity is

CCL(H, P/σ2) = maxPj ,

∑Pj≤P

r∑j=1

log2

(1 +

λ2jPj

σ2

)bps/Hz. (1.15)

The optimal power allocation depends on the SNR P/σ2 and can be solved

using an algorithm known as waterfilling. We note that the SIMO capacity

(1.9) and MISO capacity (1.12) are special cases of (1.15) where r = 1.

The MIMO capacity (1.15) can be achieved using the technique shown

in Figure 1.4. The data stream is multiplexed into r substreams, and each

substream is encoded with a capacity-achieving code corresponding to its

maximum achievable data rate (1.14). These streams are precoded and trans-

mitted over M antennas. At the receiver, the combiner is applied to yield r

substreams signals. These are independently demodulated and decoded, and

the estimated data bits are demultiplexed to reconstruct the original infor-

mation data stream.

It can be shown that at very low SNRs, it is optimal according to the

waterfilling algorithm to transmit with full power P on the dominant eigen-

mode, which is the one corresponding to the largest eigenvalue maxj λ2j . In

other words, power is transmitted on only one of the r subchannels. From

(1.15) and (1.11), the resulting capacity is approximately

log2

(1 + max

jλ2j

P

σ2

)≈ max

jλ2j

P

σ2log2 e. (1.16)


The largest eigenvalue value reflects the amplitude boost on the dominant

eigenmode due to precoding and combining. Therefore increasing the number

of antennas increases the capacity through the resulting power gain.

At very high SNRs, it is optimal to allocate equal power P/r on each of

the r subchannels. Using (1.15), the resulting capacity is approximately

r∑j=1

log2

(1 +

λ2jP

rσ2

)≈ min(M,N) log2

P

σ2, (1.17)

which follows from (1.10). The factor min(M,N) in (1.17) is known as the

multiplexing gain and indicates the number of interference-free subchannels

resulting from the decomposition of H. It is also known as the pre-log factor

and the spatial degrees of freedom.

As a result of the multiplexing gain, the capacity increases linearly as

the number of antennas increases. Therefore for a fixed transmit power and

bandwidth at high SNR, doubling the number of transmit and receive an-

tennas results in a doubling of the capacity. In order to achieve higher data

rates using fixed bandwidth resources and without using additional antennas

(for example, over a SISO channel), the transmit power needs to increase

exponentially to support a linear gain in capacity at high SNRs (1.6). This

solution would be impractical due to the prohibitive cost of larger amplifiers,

and possibly unlawful with regard to transmission regulations.

mux

Coding, modulation

Coding, modulation

Demod,decoding

Demod,decoding

de-mux

Data bits l l CombiningPrecoding�

r streams

�

r streams

Fig. 1.4 The closed-loop MIMO capacity can be achieved by decomposing the SU-MIMO

channel H into r = min(M,N) parallel subchannels (eigenmodes) using appropriate pre-

coding and combining matrices.

12 1 Introduction

1.1.3 Multiuser capacity metrics

For single-user MIMO channels, reliable communication can be achieved for

any rate less than the capacity C. For a K-user multiuser MIMO channel,

the appropriate performance limit is the K-dimensional capacity region con-

sisting of all the vectors of rates R := (R1, . . . , RK) that can be achieved

simultaneously by the K users (here, Rk ≥ 0 is the rate corresponding to

user k). The MAC capacity region CMAC is a function of the users’ channels

Hk, k = 1, . . . ,K and SNRs Pk/σ2, k = 1, . . . ,K:

CMAC(H1, . . . ,HK , P1/σ

2, . . . , PK/σ2), (1.18)

and the BC capacity region is a function of the channels HHk , k = 1, . . . ,K

and the SNR P/σ2:

CBC(HH

1 , . . . ,HHK , P/σ2

). (1.19)

To illustrate the characteristics of the multiuser capacity metrics, we con-

sider the generic two-user capacity regions shown in Figure 1.5. In general,

the rate regions for the MAC and BC are convex regions. Along the boundary

of the regions in Figure 1.5, it is not possible to increase the rates of both

users simultaneously because as the rate of one user increases, the rate of the

other user must decrease. This tradeoff occurs because the two users must

share common resources.

Given the capacity region for a K-user MU-MIMO channel, it is useful

to have a scalar metric that indicates a single “best” rate vector belonging

to the capacity region. For example, assuming that all users pay the same

per bit of information communicated, the highest revenue would be achieved

by maximizing the sum of the rates delivered to all the users. This sum-rate

capacity (or sum-capacity) metric is defined as:

maxR∈C

K∑k=1

Rk, (1.20)

where C denotes either the BC or MAC capacity region. From the perspective

of a cellular base station, the sum-capacity is useful because it measures the

maximum downlink throughput transmitted by the base over a BC or the

maximum uplink throughput over a MAC.

In Figure 1.5, the sum-capacity is achieved by the rate vector correspond-

ing to the point C on the capacity region boundary at which the tangent line


has slope −1.

(bps/Hz)

(bps

/Hz)

TDMA capacity regionMultiuser capacity region

Slope = -1

Sum-capacity rate vector

A

C

B

Fig. 1.5 Generic 2-user capacity region (MAC or BC). Sum-capacity is achieved at the

point C on the boundary. Also shown is the TDMA rate region, corresponding to time-

multiplexing the users.

Multiple-access channel capacity region

Suppose we regard Figure 1.5 as representing the multiple-access channel

capacity region. Then point A corresponds to having user 1 alone transmit

at maximum power and having user 2 be silent (and vice versa for point B).

Any rate vector lying on the line segment joining A and B can be achieved by

time-multiplexing the transmissions of user 1 and user 2 (with each point on

that line corresponding to a different time split between them). The triangular

region bounded by the axes and the line segment joining A and B is therefore

known as the time-division multiple-access (TDMA) rate region.

In general, there are rate vectors belonging to the MAC capacity region

that lie outside the TDMA rate region, and these are achieved through spatial

multiplexing by having the users transmit simultaneously over the same fre-

quency resources and jointly decoding them. The spatial multiplexing of sig-

nals from multiple users is known as space-division multiple-access (SDMA).

For the case of a single mobile antenna (N = 1), the multiplexing of users is

implemented by having each user transmit independently encoded and mod-

ulated data signals simultaneously, as shown in the top half of Figure 1.6.

Unlike the closed-loop strategy where precoding and combining creates

parallel subchannels, each user’s signal is received in the presence of inter-

ference from the other users. To account for the interference optimally, the

14 1 Introduction

receiver uses multiuser detection to jointly detect the spatially multiplexed

signals. Specifically, the receiver is based on a minimum mean-squared error

(MMSE) combiner followed by a successive interference canceller (SIC). The

SIC decodes and cancels substreams in an ordered manner so each user’s

signal is received in the presence of interference from those only the users

yet to be decoded. The MMSE-SIC architecture is more complex than the

combiner and independent decoders used for achieving closed-loop MIMO

capacity (Figure 1.4), but this is the price we pay for achieving spatial mul-

tiplexing if the transmitting antennas cannot cooperate.

At very low SNRs, the interference power is dominated by the thermal

noise power. In this regime, using an MRC is optimal, and the capacity

achieved by the kth user, averaged over i.i.d. Rayleigh realizations of the

channel hk, is approximately

NPk

σ2log2 e, (1.21)

where N = E(‖h‖2) represents the average combining gain achieved from

the multiple receive antennas. To maximize the achievable sum rate, each

user transmits with full power (Pk for user k), and the resulting average

sum-capacity at very low SNR is

NP

σ2log2 e, (1.22)

where we use P to denote the total power of the K users: P =∑K

k=1 Pk.

The sum-capacity does not depend on how the power is distributed among

the users or even how many users there are, as long as the total power is

fixed. Therefore at low SNR, the N antennas provide a power gain, but the

sum-capacity is independent of the number of users K for a fixed total power

P .

At very high SNRs, the sum-capacity of the MAC is approximately

min(K,M) log2P

σ2, (1.23)

implying that min(K,M) interference-free links can be established over the

MAC. This capacity can be achieved using the MMSE-SIC to disentangle the

users’ signals. The multiplexing gain of min(K,M) in (1.23) is the same as

that of a (K,M) SU-MIMO link with CSI and precoding at the transmitter

(1.17). It follows that full multiplexing gain could have been achieved over


the SU-MIMO link with an MMSE-SIC and without CSI and transmitter

precoding.

M

Multiple- access channel

Broadcast channel

M

User 1 Coding,

modulation

User s : Coding,

modulation

M

M

User 1Pre-

coding

User 1 :Pre-

codingM

M

Σ

User 1 Demod,decoding

User : Demod,decoding

Joint demod,

decoding:MMSE-

SIC

Joint coding,

modulation:DPC

User 1data bits

User data bits

User 1data bits

User data bits

User 1estimateddata bits

User estimateddata bits

User 1estimateddata bits

User estimateddata bits

Fig. 1.6 Strategies for achieving rate vectors on the capacity region boundary (N = 1).

For the MAC, users transmit signals simultaneously and are jointly decoded using MMSE-

SIC. For the BC, users’ data streams are jointly encoded using dirty paper coding and

then precoded.

Broadcast channel capacity region

If we now regard Figure 1.5 as the broadcast channel capacity region, then

point A (resp. B) corresponds to transmitting exclusively to user 1 (resp. user

2) at full power. Again, points on the line segment joining A and B are

achievable by time-sharing the channel between the users, and the triangular

region bounded by the axes and the A-B line segment is called the time-

division multiple-access (TDMA) rate region.

As was the case for the MAC, rate vectors on the capacity region boundary

of the BC are achieved by spatially multiplexing signals for multiple users.

One method for multiplexing is linear precoding as shown in Figure 1.7, where

the users’ data streams are independently encoded and transmitted simulta-

neously using different linear precoding vectors. The precoding vectors are a

16 1 Introduction

function of the channel realizations hH1 , . . . ,hH

K , and hence CSI is required

at the transmitter. For example, MRT precoding could be implemented for

each user. (If the antennas are highly correlated, each MRT vector creates

a directional beam pattern pointing in the direction of its desired user as

shown in Figure 1.7.) Because the MRT vector is determined myopically for

each user and does not account for the interaction between beams, each user

would experience interference from signals intended for the other users.

In general, SDMA implemented with linear precoding alone is not optimal

in that it does not achieve rate vectors lying on the BC capacity region

boundary. The optimal technique uses linear precoding preceded by a joint

encoding and modulation technique known as dirty paper coding (DPC). The

DPC encoder output provides data symbols for each of the K users which are

precoded and transmitted simultaneously. DPC uses knowledge of the users’

channels and data streams to perform coding in an ordered fashion among the

users, and thereby to remove interference at the transmitter. This is similar

to the effect of the SIC which removes the interference at the receiver through

ordered decoding. The DPC and precoders are designed so that each user’s

received signal does not experience interference from users encoded after it.

At very low SNRs, the BC capacity region becomes equivalent to the

corresponding TDMA capacity region, and SDMA no longer provides any

benefit over simple single-user transmission. In this regime, the sum rate is

maximized by serving the single user whose MISO channel yields the highest

capacity. From (1.12) and (1.11), the resulting sum-capacity at asymptotically

low SNRs is approximately

maxk=1,...,K

‖hk‖2 P

σ2log2 e. (1.24)

Therefore the multiple transmit antennas provide power gain through pre-

coding, and additional users increase the capacity according to the statistics

of maxk=1,...,K |hk|2.At high SNRs, the sum-capacity of the BC is the same as the MAC’s (1.23)

if the BC power P is the same as the total power of the MAC users:

min(K,M) log2P

σ2. (1.25)

The multiplexing gain min(K,M) achieved over the BC requires CSI at the

transmitter but does not require coordination between the users. Therefore a


(K,M) SU-MIMO channel could achieve the same multiplexing gain without

jointly processing the received signals across the antennas.

C.M

User 1Pre-

coding

User 1 Pre-

codingM

M

Σ

User 1Coding,

modulation

User : Coding,

modulation

MM

User 1data bits

User data bits

Fig. 1.7 Suboptimal spatial multiplexing for the BC can be implemented as spatial di-

vision multiple-access (SDMA) by simultaneously transmitting multiple precoded data

streams. If the antennas are closely spaced and highly correlated, precoding creates direc-

tional beams.

1.1.4 MIMO performance gains

To illustrate the performance gains of the multiple antenna techniques, we

compare the capacity performance over the following channels:

• (1,1) SU-SISO with channel h ∈ C and power constraint P

• (4,4) SU-MIMO with channel H ∈ C4×4 and power constraint P

• ((4,1),4) MU MAC with channel hk ∈ C4×1 and power constraint Pk =

P/4 for each user k = 1, . . . , 4

• (4,(4,1)) MU BC with channel hHk ∈ C

1×4 for each user k = 1, . . . , 4 and

power constraint P

The channels are assumed to all be i.i.d. Rayleigh, and the noise is assumed to

have variance σ2 so the average SNR in all cases is P/σ2. For the single-user

channels, we consider the SISO capacity (1.6) and closed-loop MIMO capacity

(1.15) averaged over the i.i.d. channel realizations. For the multiuser channels,

we consider the sum-capacity averaged over the i.i.d. channel realizations.

To better observe the multiplexing gains at high SNR and power gains at

low SNR, the capacity performances are illustrated in Figures 1.8 and 1.9 for

18 1 Introduction

the range of high and low SNRs, respectively.

Capacity at high SNR

At high SNRs, multiple antennas provide capacity gains through spatial mul-

tiplexing. For the SU channels, the multiplexing gain min(M,N) (1.17) gives

the slope of the capacity with respect to log2 P for asymptotically high SNR.

The capacity curves for SISO and MIMO are parallel, respectively, to the

lines given by log2(P/σ2) and 4 log2(P/σ

2). For every increase of SNR by

3 dB, the MIMO capacity increases by 4 bps/Hz and the SISO capacity in-

creases by 1 bps/Hz. While the scaling in (1.17) applies to asymptotically

high SNR, the asymptotic behavior is already apparent in Figure 1.8 for an

SNR of 15 dB. Increasing the number of antennas results in a linear capacity

increase, whereas increasing the transmit power results in only a logarithmic

increase. Therefore achieving very high data rates using SISO becomes infea-

sible due to the required power. For example, at P/σ2 = 15 dB, the MIMO

capacity is about 16 bps/Hz. Achieving the same capacity with SISO requires

an SNR of 48 dB, so MIMO saves more than three orders of magnitude of

power. The significant capacity gains and power savings provided by spatial

multiplexing are the key benefits that make MIMO techniques so appealing

at high SNR.

We can also compare the SU-MIMO channel performance with the BC and

MAC performance. Because the multiplexing gain of the MU-MIMO chan-

nels is min(K,M) = 4, the sum-capacity slope with respect to SNR is the

same as the SU-MIMO channel. Figure 1.8 shows that the absolute capacities

(and not just their slopes) are nearly identical, indicating that at high SNR,

little is gained by coordinating the transmission or reception among multiple

single-antenna users.

Capacity at low SNR

For the SU-MIMO channel at low SNR, transmitting a single stream is opti-

mal. In this regime, a factor of α power gain in the SNR results in a capacity

gain of α. In some cases, these gains can be larger than those attained from

multiplexing at high SNR. For example, as highlighted in Figure 1.9, at SNR

= -15 dB, the power gain achieved from precoding and receiver combining

over the (4,4) SU-MIMO channel results in a factor of 9 gain in capacity com-

pared to SU-SISO. Even though it is no longer optimal to transmit a single

stream at higher SNRs, the capacity gain of MIMO versus SISO is greater

than 4 in the range of SNRs shown in this figure.

1.2 Overview of cellular networks 19

At low SNRs there is a significant difference between the capacity of the

SU-MIMO channel and the sum-capacity of either the MAC or BC. It follows

that in the regime of low SNR, coordination at both the transmitter and

receiver, respectively in the form of precoding and combining, provides benefit

over having coordination at just one end or the other.

1.2 Overview of cellular networks

The previous section describes the capacity performance of narrowband

MIMO channels that model the communication links in a cell between a

single base and its assigned user(s). In a cellular network consisting of mul-

tiple bases and multiple assigned users per base, the transmissions from one

cell may cause co-channel interference at the receivers of other cells, resulting

in degraded performance. MIMO links form the basis of the cellular network’s

underlying physical layer, and in order to efficiently leverage the benefits of

multiple antennas, it is necessary to understand the system-level aspects of

the network architecture, the nature of co-channel interference, and how the

interference can be mitigated [2].

1.2.1 System characteristics

In this section we describe the system-level aspects of cellular networks that

are relevant for modeling physical-layer performance. These include the par-

titioning of the geographic region into cells and sectors, the technique for

supporting multiple users in the network, and the allocation of spectral re-

sources in packet-based systems.

Cell sites and sectorization

In practice, the location of base stations depends on factors such as the

user distribution, terrain, and zoning restrictions. However for the purpose

of simulations, it is convenient to assume a hexagonal grid of cells where a

base is located at the center of each cell, as shown in Figure 1.10. Each user

is assigned to the base with the best radio link quality which, as a result of

channel shadow fading, may not necessarily be the one that is closest to it.

A user lying in a particular hexagonal cell is not necessarily assigned to the

20 1 Introduction

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

SNR (dB)

Ave

rage

cap

acity

and

sum

-cap

acity

(bps

/Hz)

(1,1) SU

(4,(4,1)) BC((4,1),4) MAC

(4,4) SU 4log2(SNR)

log2(SNR)

Fig. 1.8 At high SNR, the average (M,N) SU-MIMO capacity scales linearly with the

number of antennas min(M,N) as a result of spatially multiplexing independent data

streams. The multiplexing gain gives the slope of the capacity with respect to log2(P/σ2)

at high SNR.

-20 -15 -10 -5 00

0.5

1

1.5

2

2.5

3

SNR (dB)

Ave

rage

cap

acity

and

sum

-cap

acity

(bps

/Hz)

(1,1) SU(4,(4,1)) BC((4,1),4) MAC

(4,4) SU

9×

7×

6×

Fig. 1.9 At low SNR, the SU-MIMO capacity gains over SU-SISO are greater compared

to the gains in the high-SNR regime. Coordinating both the transmit and receive antennas

(as in the SU-MIMO channel) provides benefits over coordination at only one end or the

other.


base at the center of that cell. The cell boundaries therefore highlight the

location of the bases but do not indicate the assigned bases.

A fundamental characteristic of cellular networks is that the spectral re-

sources are reused at different cells in order to improve the spectral efficiency.

In general, the channel bandwidth is partitioned into subbands and these are

assigned to different cells to trade off between the harmful effects of interfer-

ence and the gains from frequency reuse. For example in early analog cellular

networks, the network was partitioned into groups of at least seven cells, and

each cell within a cluster was assigned a set of subbands which did not inter-

fere with those assigned to other cells in the cluster. This concept of reduced

frequency reuse is shown in the left subfigure of Figure 1.11. A user assigned

to a particular cell received interference from cells in adjacent clusters, but

the interference power was low enough to ensure reliable demodulation of the

desired signal. With the introduction of digital communication techniques,

reliable demodulation could be achieved in the presence of higher interfer-

ence power. In contemporary cellular networks, the entire channel bandwidth

is reused at each cell under universal frequency reuse. Compared to a system

with reduced frequency reuse, users assigned to a particular cell experience

more interference, but each cell uses more bandwidth so the overall system

spectral efficiency is potentially higher.

At each cell site, the spatial characteristics of the channels are determined

by the base station antenna architecture and the transceiver technique. Fig-

ure 1.12 shows five examples of different architectures. Column A shows a

single omni-directional antenna for the base. An omni-directional antenna can

be implemented as a dipole element that has a circular cross-section. Mul-

tiple antenna techniques could be implemented using three omni-directional

antennas at the base, as shown in Column B.

Multiple antennas at each site could also be used for sectorization, which

is the radially partitioning of a cell site into multiple spatial channels. In

practice, cell sites are often partitioned into three sectors, as shown in the

right subfigure of Figure 1.10 and implemented using the antenna architecture

in column C of Figure 1.12. Each antenna element has a rectangular cross-

section and uses a physical reflector to focus energy in a beam towards the

desired direction. Sectorization is an efficient technique for achieving higher

spectral efficiency if the bandwidth is reused at each sector and if the inter-

ference between sectors is tolerable. MIMO techniques can be implemented

in a sector by deploying multiple directional antennas, as shown in column

D. If the antennas are closely spaced, directional beams (Figure 1.7) can be

22 1 Introduction

formed from each array. In column E, two closely-spaced directional antennas

are deployed in each sector, and these antenna pairs each form two directional

beams. From the perspective of the user, each beam is indistinguishable from

the antenna pattern of a sector served by a single directional antenna. There-

fore a total of six sectors, each with M = 1 effective antenna, can be formed

at the site using this antenna configuration.

……

…

…

…

…

………

…

…

…

sector per site sectors per site

Fig. 1.10 A hexagonal grid of cells will be used for performance simulations. On the left,

the antennas at each base are omni-directional. On the right, each cell site is partitioned

radially into three sectors.

Multiple access

In order to support multiple users in a cellular network, a cellular standard

defines a multiple-access technique to specify how the spectral resources are

shared among the users assigned to a cell or sector. These techniques include

frequency division multiple-access (FDMA), time division multiple-access

(TDMA), code division multiple-access (CDMA), and orthogonal frequency

division multiple-access (OFDMA) [3].

In first-generation analog cellular networks based on FDMA, the channel

bandwidth is partitioned into subcarriers, and a user is assigned to a single

subcarrier. Subcarriers can be reused at different cells if they are sufficiently

far apart that the interference power is negligible. TDMA builds on FDMA by

imposing a slotted time structure on each subchannel. By assigning users to

different time slots, multiple users can share each subcarrier, thereby increas-


Universal frequency reuseReduced frequency reuse

f1f1 f2 f3

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4

f1

f6

f7

f2

f5

f3

f4f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f4 f5 f6 f7

Fig. 1.11 Under universal frequency reuse, cells operate on the same frequency. Under

reduced frequency reuse, cells operate on a fixed reuse pattern to mitigate interference.

Number of sectors per cell site ( )

Number of antennas persector ( )

B C D EA

1

1

1

3

3

1

3

2

6

1

Fig. 1.12 Examples of different antenna configurations for a cell site.

24 1 Introduction

ing the spectral efficiency. For example, slots can be allocated in a round-robin

fashion among three users such that each user accesses the subcarrier on every

third slot. Under CDMA with direct sequence spread spectrum, users access

the channel with different code sequences. These sequences are designed to

have low cross-correlation so that interference is minimized.

The latest cellular standards use orthogonal frequency division multiple-

access (OFDMA). Here, as in FDMA, the bandwidth is partitioned into mul-

tiple subcarriers. However unlike FDMA, where a guard band is inserted

between subcarriers to prevent inter-carrier interference, OFDMA improves

the spectral efficiency by using an inverse Fast Fourier Transform (FFT) to

multiplex signals across multiple subcarriers. As a result, signals overlap in

the frequency domain yet are mutually orthogonal.

The 3GPP Long-term evolution (LTE) system [4] is an example of a cur-

rent cellular standard that uses OFDMA. It can support channel bandwidths

up to 20 MHz and partitions it into subcarriers of bandwidth 15 kHz. The

time domain consists of 1 ms subframes spanning 14 symbol periods of du-

ration 0.07 ms. The minimum unit of resource allocation is known as a phys-

ical resource block (PRB), which consists of 12 contiguous subcarriers (180

kHz bandwidth) in the frequency domain and 1 subframe in the time do-

main. The time and frequency resources for OFDMA can be represented as

a two-dimensional grid as illustrated in Figure 1.13, where each segment cor-

responds to a PRB. Another OFDMA-based standard that uses a similar

structure but with different parameters and terminology is IEEE 802.16e [5].

For simplicity we will use the LTE terminology in our discussion.

Scheduling and resource allocation

First- and second-generation cellular networks typically used circuit-switched

communication to provide voice service over constant bit rate connections. In

contrast, contemporary packet-based systems such as LTE and IEEE 802.16e

employ packet switching where all traffic, regardless of content or type, is

packaged as blocks of data known as packets. Each base (sector) employs

a scheduler that allocates spectral resources for packet transmission among

its assigned users. A separate scheduler is used for the uplink and downlink,

and an independent allocation of resources can occur as often as once every

subframe. As shown in Figure 1.13, the allocation of the resources is dynamic

such that each PRB or group of PRBs can be allocated to a different user

on each subframe. An encoding block for a given user occurs over a block


…time

frequ

ency

User 2 MU-MIMOUsers 2, 3

User 1 SU-MIMOUser 4

…

1ms

150kHz

Fig. 1.13 Under the LTE standard, OFDMA is used, and the time/frequency resources

are partitioned into physical resource blocks with bandwidth 180 kHz and 1 ms subframe

time intervals. A scheduler allocates resources among users. MIMO techniques can be

implemented for each resource block. For example, on the rightmost subframe, user 4 is

allocated 3 resource blocks with each supporting SU-MIMO spatial multiplexing. Users 2

and 3 are spatially multiplexed on each of the lower resource blocks.

of symbols that spans one subframe in the time domain and one or more

resource blocks in the frequency domain.

The scheduler allocates resources in order to maximize the total through-

put while meeting quality of service (QoS) requirements for each user’s appli-

cation. These requirements include the average data rate and the maximum

tolerable latency. For example, a large file transfer would require a large rate

and could tolerate a large latency whereas a streaming audio application

would require a lower rate and smaller latency.

For each transmission, the scheduler determines the appropriate trans-

mission rate, which is based on measured channel characteristics and QoS

requirements, and characterized by the coding rate and symbol modulation.

This technique is known as rate adaptation. For downlink data transmission,

a downlink control channel transmitted on resources orthogonal to the data

traffic channel notifies the users of the modulation and coding scheme. For up-

link data transmission, the downlink control channel notifies the scheduled

users which resources to use and what the modulation and coding scheme

should be.

26 1 Introduction

In summary, the cellular infrastructure consists of cell sites that are par-

titioned into sectors. For a given sector, a base station communicates with

multiple users and the sharing of spectral resources between the users is de-

fined by the multiple-access technique. In contemporary networks, multiple-

access is based on OFDMA, where the channel bandwidth is partitioned into

narrowband subcarriers and time is partitioned into subframes. During each

subframe, a base station scheduler allocates contiguous blocks of subcarriers

(PRBs) to different users to optimize its performance metric.

If multiple antennas are available at the base and to mobile users, spatial

multiplexing can be implemented over the PRBs. As shown in the rightmost

subframe of Figure 1.13, user 4 is allocated 3 PRBs, and SU-MIMO spatial

multiplexing is implemented for each PRB. Equation (1.1) can be used to

model the received signal for either the uplink or downlink. Users 2 and 3 are

spatially multiplexed on each of the lower 3 PRBs. If the transmission occurs

for the uplink, equation (1.2) can be used to model the received signal; for

the downlink, equation (1.3) can be used.

1.2.2 Co-channel interference

In the discussion of SU and MU-MIMO channel capacity in Section 1.1, we as-

sumed that the links were corrupted by additive noise. The performance was

characterized by the signal-to-noise ratio. In cellular systems, the link perfor-

mance is dependent on the resource allocation between bases and is affected

by co-channel interference arising from transmissions occuring on common

frequency channels. If we treat the interference as an additional source of

noise, the performance can be characterized by the signal-to-interference-

plus-noise ratio (SINR).

As shown in Figure 1.14, interference could occur between cells and be-

tween sectors belonging to the same cell site. On the uplink, a base could

receive intercell interference from co-channel users assigned to other bases,

and on the downlink, a user could receive intercell interference from bases not

assigned to it. Intracell interference could occur between sectors belonging to

the same cell site as a result of power leaking through non-ideal sector side-

lobes. Within a sector, interference could occur as a result of multiuser spatial

multiplexing if the spatial channels are not orthogonal. Other than spatial

interference, we assume there is no other intrasector interference within a


given time-frequency resource block.

Downlink Uplink

Desired signalInterfering signal

Inte

rcel

lin

terfe

renc

eIn

trace

llin

terfe

renc

e

Fig. 1.14 Intercell interference occurs between cells. Intracell interference occurs between

sectors belonging to the same cell site. Intrasector interference occurs between spatially

multiplexed users belonging to the same sector.

SINR and geometry

For the SU-MIMO channel in Section 1.1, we made a distinction between

the channel SNR, defined as P/σ2, and the SNR at the input of the decoder

which is a function of P/σ2, the channel realization H, and the transceiver

technique which defines the precoding and combining. For a cellular network,

the SINR at the input of the decoder is a function of the transceivers and

the transmit powers and channel realizations of the desired and interfering

sources. Analogous to the channel SNR for the SU-MIMO channel, we define

the geometry as the average received power from the desired source divided by

the sum of the average received noise and interference power. The geometry

accounts for the distance-based pathlosses and shadow fading realizations but

is independent of the number of antennas. For example, on the downlink, a

user at the edge of a cell is said to have low geometry if the average power

received from its serving base is low compared to the co-channel interference

power received from all other bases. On the other hand, a user very close to

28 1 Introduction

its assigned base has a high geometry if its received power from the desired

base is high compared to the interference power.

As we will see, in typical cellular networks, the range of the geometry is

about -10 dB to 20 dB for both the uplink and downlink. We can interpret

the performance results in Figures 1.8 and 1.9 in a cellular context by rela-

beling the channel SNR on the x-axis with the geometry. For the SU-MIMO

capacity performance, a geometry of 20 dB corresponds to a user positioned

very close to its serving base. For the MU-MIMO sum-capacity performance,

a geometry of 20 dB corresponds to all four users positioned very close to

their serving base.

Impact of interference on capacity

Suppose that a user on the downlink receives average power P watts from

its desired base and average total power αP (0 ≤ α < 1) from the interfering

bases. In the presence of additive Gaussian noise power σ2, the geometry

is P/(σ2 + αP ). Assuming there is no Rayleigh fading, the corresponding

capacity (in bps/Hz) is

log2

(1 +

P

σ2 + αP

). (1.26)

With α fixed, increasing the transmit power of both the desired and interfer-

ing bases by β watts results in a higher capacity because

P + β

σ2 + α(P + β)>

P

σ2 + αP(1.27)

for any β > 0. As P approaches infinity, the capacity approaches the limit

log2(1 + 1/α) for α > 0.

If the noise power is significantly greater than the interference power (i.e.,

αP � σ2), the performance is said to be noise-limited. In this case, increasing

P results in a linear capacity gain if P/σ2 is small and a logarithmic gain

if P/σ2 is large. On the other hand, if the interference power is significantly

greater than the noise power (i.e., αP � σ2), the performance is interference-

limited. As P increases, the capacity approaches its limit of log2(1 + 1/α).

Figure 1.15 shows an example of link capacity (1.26) versus SNR P/σ2

for α = 0, 0.1, 0.5. When there is no interference (α = 0), the performance

is noise limited for the entire range of SNR. For α = 0.1, the performance is

noise limited for the lower range of SNR and interference limited for SNRs

greater than 20 dB. For α = 0.5, the performance is interference-limited for


SNRs greater than 10 dB. It is desirable to operate a system so that the SNR

is at the transition point between noise and interference-limited performance.

At this point (for example, operating at an SNR of 10 dB for α = 0.5), the

power is used efficiently and increasing P further would not result in signifi-

cant performance gains. If on the other hand P is fixed but interference can

be mitigated to reduce α, significant performance gains can be achieved in

the interference-limited SNR regime. For example in Figure 1.15, the gains

from reducing α from 0.5 to 0.1 are significant at 20 dB SNR but insignificant

at 0 dB SNR.

-10 -5 0 5 10 15 20 25 300

2

4

6

8

10

P/σ2 (dB)

Cap

acity

(bps

/Hz) α = 0

α = 0.1

α = 0.5

Fig. 1.15 Link capacity (1.26) versus SNR P/σ2 with interference scaling factor α. If

α = 0, the performance is noise-limited, and the capacity increases without bound as SNR

increases. If α > 0, the performance is interference limited, and the capacity saturates as

SNR increases.

Interference mitigation techniques

As discussed in Section 1.2.1, reduced frequency reuse can mitigate inter-

ference between cells by ensuring adjacent cells are assigned to orthogonal

subcarriers. Other techniques for interference mitigation include the follow-

ing.

30 1 Introduction

• Power control. Transmit powers could be adjusted to meet target SINR

requirements. Transmitters closer to their intended receiver could achieve

their target with less power, resulting in less interference caused to other

cells. Power control is often used in circuit-based systems to ensure the

SINR is high enough to support reliable voice communication.

• Interference averaging. In CDMA standards, transmissions are spread

over a wide bandwidth using pseudo-random code spreading or frequency

hopping. The resulting interference power is averaged over the wider band-

width, resulting in interference statistics with lower variance.

• Soft handoff. Interference can be reduced by allowing a terminal to com-

municate simultaneously with two or more bases over common spectral

resources. On the downlink, adjacent bases send the same information to

the user, and these signals can be combined to improve reliability. On the

uplink, a similar diversity advantage can be achieved by detecting a user’s

signal independently at multiple bases and having a central controller de-

cide which signal is the most reliable. Soft handoff is spectrally inefficient

because the resources could have been used to serve additional users.

These three mitigation techniques can be implemented using a single an-

tenna at the base and mobile. Using multiple antennas, interference can be

mitigated by exploiting the spatial domain. The transmitter precoding tech-

niques described in Section 1.1 create spatial channels that focus the beam

towards the desired user while implicitly reducing the interference to any

other receiver lying outside the spatial channel. If, on the downlink, a base

has knowledge of the CSI of users assigned to other bases, it can intentionally

steer transmissions away from those users to mitigate interference. Similarly,

if on the uplink a base has knowledge of the CSI from interfering users, it can

mitigate their interference through judicious receiver combining. In essence,

multiuser detection is applied by a base jointly to assigned and interfering

users. Because the MIMO techniques can create only a limited number of

interference-free spatial channels, an important design decision for optimiz-

ing system performance is to determine the allocation of spatial resources

between the assigned and interfering users. Assigning more resources to the

assigned users potentially increases the multiplexing gain but the SINR of

each user could be lower as a result.

The performance of interference mitigation techniques can be improved

by coordinating the transmission and reception of base stations. Coordina-

tion strategies require enhanced baseband processing and additional backhaul

1.3 Overview of the book 31

network resources for sharing information between the bases, and the per-

formance gains from coordination must be weighed against the cost of the

added complexity. Spatial interference mitigation will be discussed in more

detail in Chapters 5 and 6.

1.3 Overview of the book

A goal of this book is to apply theoretical Shannon capacity results to the

practical performance of contemporary cellular networks. We do so by ad-

dressing the central problem of designing a MIMO cellular network to maxi-

mize the throughput spectral efficiency per unit area. Our method is to de-

velop an understanding of MIMO-channel capacity metrics and to evaluate

them in a simplified simulation environment that captures key features of real-

world networks that are most relevant to the physical-layer performance. The

simulation environment is not specific to a particular standard and is broad

enough to capture characteristics of general contemporary packet-based net-

works.

During the course of this book, we hope to illuminate general principles for

MIMO communication and wireless system design that can also be applied to

networks beyond the cellular realm. Some of the questions relating to MIMO

performance include the following. How does the capacity of a MIMO link

scale with the number of antennas at both ends, in various SNR regimes?

What transmitter and receiver architectures are required to achieve capac-

ity? What is the impact of channel state information at the transmitter? In

multiuser MIMO channels, how does the sum-capacity scale with the number

of antennas or the number of users? How far from optimal are the simpler

transmitter and receiver architectures that are more suitable for practical

implementation?

Further, in the context of a cellular network, what is the best way to

use multiple antennas, keeping the impact of intercell interference in mind?

How can multiple antennas be used to mitigate intercell interference? When is

single-user MIMO preferable to multiuser MIMO? How do MIMO techniques

compare to simpler alternatives for improving area spectral efficiency such as

sectorization?

The book is divided into two main parts. The first part consists of Chapters

2 through 4 and covers the fundamentals of communication over SU- and MU-

32 1 Introduction

MIMO channels. The second part consists of Chapters 5 through 7 and applies

the fundamental techniques in the design and operation of cellular networks.

Figure 1.16 gives an overview of the chapter contents, channel models, and

metrics.

• Chapter 2 discusses the capacity of the SU-MIMO channel. We describe

techniques for achieving capacity and also suboptimal techniques including

linear receivers, space-time coding, and limited-feedback precoding.

• Chapter 3 focuses on optimal techniques for MU-MIMO channels. The

capacity regions and capacity-achieving techniques for the MAC and BC

are described. Concepts useful for numerical performance evaluation are

introduced, including a duality relating the BC and MAC capacity re-

gions, and the numerical techniques for maximizing sum-rate performance

metrics. Asymptotic sum-rate performance results provide insights into

how the optimal MU-MIMO techniques should be implemented in cellular

networks.

• Chapter 4 describes suboptimal techniques for the MAC and BC. Due

to the challenges of capacity-achieving techniques for the BC, most of the

chapter focuses on various suboptimal BC precoding techniques.

• Chapter 5 shifts the discussion from isolated MIMO channels to cellu-

lar networks, and develops a general simulation methodology for evaluat-

ing cellular network performance that is applicable to any contemporary

packet-switched cellular standard. This chapter also describes practical as-

pects of these networks including scheduling and procedures for acquiring

channel state information.

• Chapter 6 presents numerical simulation results for a variety of network

architectures. It consists of three sections: single-antenna bases, multiple-

antenna bases, and coordinated bases. This chapter synthesizes material

from the previous chapters and concludes with a flowchart that gives design

recommendations for applying MIMO in cellular networks.

• Chapter 7 describes specific multiple-antenna techniques for existing and

future cellular standards. We consider how the general principles and tech-

niques discussed in the other chapters are specifically implemented in these

standards.

1.3 Overview of the book 33

mul

tiacc

ess

chan

nel (

MA

C)

Met

rics

Cap

acity

Cap

acity

regi

ons

Sum

-rat

e ca

paci

ties

MIM

O c

hann

el

Prop

ortio

nal-f

air c

riter

iaM

ean

thro

ughp

ut p

er s

iteP

eak

user

rate

Cel

l-edg

e us

er ra

te

Equa

l-rat

e cr

iteria

Thro

ughp

ut p

er s

ite

Upl

ink

chan

nel,

base

bre

ceiv

ed s

igna

l

Dow

nlin

k ch

anne

l, us

er k

rece

ived

sig

nal

Cha

nnel

mod

elC

hapt

er

2 3, 4

5, 6

�

�

Cha

nnel

ty

pe

broa

dcas

t cha

nnel

(BC

)

Sin

gle-

user

M

IMO

Mul

tiuse

rM

IMO

Cel

lula

rne

twor

k

Fig. 1.16 Overview of chapter contents, channel models, and metrics.

Chapter 2

Single-user MIMO

In this chapter we study the single-user MIMO channel which is used to

model the communication link between a base station and a user. We explore

in more detail the fundamental results that were briefly described in the

previous chapter. We derive the open-loop and closed-loop MIMO channel

capacities and describe techniques for achieving capacity including architec-

tures known as V-BLAST and D-BLAST. We also describe classes of subop-

timal techniques such as linear receivers, space-time coding for transmitting

a single data stream from multiple antennas, and precoding when there is

limited knowledge of CSI at the transmitter.

2.1 Channel model

Figure 2.1 shows the baseband model for a single-user (M,N) MIMO link

with M transmit and N receive antennas. A stream of data bits is com-

municated over the channel. We let d(i) ∈ {+1,−1} represent the data

bit with index i = 0, 1, . . .. The data stream is processed to create a se-

quence of transmitted data symbols. We let s(t)m ∈ C denote the complex

baseband signal transmitted from antenna m during period t. For a given

symbol period t, the channel between the mth (m = 1, . . . ,M) transmit an-

tenna and the jth (j = 1, . . . , N) receive antenna is characterized by a scalar

value h(t)j,m ∈ C which represents the complex amplitude of the narrowband,

frequency-nonselective channel. Because each receive antenna is exposed to

all transmit antennas, the baseband signal received at antenna j during time

t can be written as a linear combination of the transmitted signals:

H. Huang et al., MIMO Communication for Cellular Networks, DOI 10.1007/978-0-387-77523-4_ , © Springer Science+Business Media, LLC 2012

352

36 2 Single-user MIMO

x(t)j =

M∑m=1

h(t)j,ms(t)m + n

(t)j , (2.1)

where n(t)j is complex additive noise. By stacking the received signals from

all N antennas in a tall vector, we can write:⎡⎢⎢⎣x(t)1...

x(t)N

⎤⎥⎥⎦ =

⎡⎢⎢⎣h(t)1,1 · · · h

(t)1,M

.... . .

...

h(t)N,1 · · · h(t)

N,M

⎤⎥⎥⎦⎡⎢⎢⎣s(t)1...

s(t)M

⎤⎥⎥⎦+

⎡⎢⎢⎣n(t)1...

n(t)N

⎤⎥⎥⎦ (2.2)

which can be written in the more compact form

x(t) = H(t)s(t) + n(t), (2.3)

where x(t) ∈ CN×1, H(t) ∈ C

N×M , s(t) ∈ CM×1, and n(t) ∈ C

N×1. For

simplicity, we will typically drop the time index t. The noise vector n

is assumed to be zero-mean, spatially white (ZMSW), circularly symmet-

ric, additive complex Gaussian, with each component having variance σ2:

E(nnH) = σ2IN , where IN is the N × N identity matrix. (When the noise

is spatially colored, i.e., when the covariance of n is not a multiple of the

identity matrix, we can suppose that the receiver whitens the noise first, by

multiplying the received signal vector by the inverse square root of the noise

covariance.) The components of the signal vectors s(t), t = 1, 2, . . . are the en-

coded symbols obtained by processing an information bit stream d(1), d(2), . . .

which we denote by{d(i)}. The signal vector is modeled as a stationary ran-

dom process with zero mean E(s) = 0M and covariance Q := E(ss)H . The

signal is subject to the power constraint trQ = E(‖s‖2) = P .

The realization H(t) is drawn from a stationary, ergodic random process to

model the fading of the wireless channel. Due to the movement of the trans-

mitter, receiver, and local scatterers, the signal transmitted from antenna m

and received by antenna j experiences multipath fading caused by varying

path lengths to the scatterers. As a result of the central limit theorem, the

complex amplitude of the combined multipath signals can be modeled as a

complex Gaussian random variable. If the spacing between the M transmit

antennas is sufficiently large relative to the channel angle spread (which is

determined by the height of the antennas relative to the height of the local

scatterers), then the M channel coefficients h(t)j,1, . . . , h

(t)j,M for receive antenna

j will be uncorrelated. Likewise, if the spacing between the N receive anten-

nas is sufficiently large, then the N channel coefficients h(t)1,m, . . . , h

(t)N,m for

2.1 Channel model 37

transmit antenna m will be uncorrelated. A channel in which the coefficients

of H(t) are uncorrelated (or weakly correlated) is said to be spatially rich.

Typically, we will assume in this book that for a given symbol interval t,

the elements ofH(t) are not only spatially rich but also independent and iden-

tically distributed (i.i.d.) complex Gaussian random variables with zero mean

and unit variance. Because the amplitude of each element has a Rayleigh dis-

tribution, this channel distribution is known as an i.i.d. Rayleigh distribution.

As a result of the channel normalization, the average received signal power

is P , and the signal-to-noise ratio (SNR), defined as the ratio of the received

signal power and noise power, is P/σ2.

With regard to the time evolution of the channel realizations, we define

two types of channel model.

1. Fast-fading: the channel changes fast enough between symbol periods

that each coding block effectively spans the entire distribution of the ran-

dom process (i.e., ergodicity holds).

2. Block-fading: the channel is fixed for the duration of a coding block, but

it changes from one block to another.

In practice, coding block lengths are on the order of a millisecond, so users

with low mobility (stationary or pedestrian users) experience slowly fad-

ing channels consistent with the block-fading model. In this book, we fo-

cus mainly on the block-fading model. Further, we usually assume an i.i.d.

Rayleigh fading model for the channel.

In the rest of this section, we briefly describe more general channel models

that account for propagation environments that are not spatially rich and

therefore induce correlated fading across transmitter and receiver antenna

pairs.

2.1.1 Analytical channel models

Analytical channel models attempt to describe the end-to-end transfer func-

tions between the transmitting and receiving antenna arrays by accounting

for physical propagation and antenna array characteristics [6]. Most analyti-

cal channel models capture the various propagation mechanisms through the

correlations of the random channel coefficients. Below we describe the most

well-known correlation-based analytical MIMO channel models.


M M Receiver processing

Transmitter processing

Fig. 2.1 The (M,N) single-user MIMO link. A stream of data bits{d(i)

}is processed

to form a stream of encoded symbol vectors{s(t)

}. The signal is transmitted from M

antennas over the channel H and is received by N antennas. The received signal{x(t)

}is

processed to provide estimates of the data stream bits{d(i)

}.

2.1.1.1 Kronecker MIMO channel model

The Kronecker MIMO channel model is probably the best-known correlation-

based model and stems from early efforts in the community [7–11] to find

models which correspond to a given pair of transmission and receiver cor-

relation matrices (denoted for simplicity as RT = E(HHH

)and RR =

E

(HHH

), respectively). It hinges on the assumption that these two corre-

lation matrices are separable. Mathematically, this assumption is expressed

as

RH = RT ⊗RR, (2.4)

where RH is defined as RH = E

(vec (H) vec (H)

H), where the vec operator

stacks the columns of the operand matrix vertically, and ⊗ denotes the Kro-

necker product between two matrices. It can be shown that, in this case, the

channel matrix can be expressed as

H = R1/2R Hi.i.d.R

1/2T (2.5)

where Hi.i.d. is a N ×M matrix of i.i.d. circularly symmetric complex Gaus-

sian random variables of zero mean and unit variance. The assumption of

separable transmit/receive correlations of course limits the generality of this

model, as it is unable to capture any coupling between direction of departure

and direction of arrival spectra. Examples of simple channel models that are

not captured by the Kronecker channel model are the so-called “keyhole chan-

nel,” as well as the single- and double-bounce models [12] described below.


However, the Kronecker model has been very popular due to its successful

role in quantifying MIMO capacity of correlated channels as well as to its

modularity, which allows separate transmitter and receiver array optimiza-

tion.

2.1.1.2 Single-bounce analytical MIMO channel model

In this case we assume that the signal transmitted from each transmitter

bounces once off each of a set of (say V ) scatterers before it reaches any re-

ceiver antenna. In this single-bounce case, the MIMO channel can be modeled

as

HSB (N,M, V ) = ΦR (N,V )Hi.i.d. (V, V )ΦHT (M,V ) , (2.6)

where ΦR and ΦT are matrices that define the electrical path lengths from

the V scatterers and the N , M antenna elements, respectively. In fact, it

turns out that these matrices are the matrix square roots of the correlation

matrices on each side of transmission, i.e.

HSB (N,M, V ) = R1/2R (N,N)Hi.i.d. (N,M)R

1/2T (M,M) . (2.7)

In the above, the first two arguments within the parentheses denote the

matrix dimensions; the third argument, when present, denotes the assumed

number of scatterers. The model in (2.7) has been used successfully to char-

acterize many practical cases where correlation among antenna elements on

each side of the link is present (e.g. due to their proximity or a limited an-

gle spread), despite the fundamental richness of the in-between propagation

environment. In other words, it models local correlation well. It should be

noted of course that when V is smaller than min (M,N), the channel in (2.7)

will suffer severe degradation in its richness, as it will lose rank (notice that

such a phenomenon cannot be captured by the Kronecker model in (2.5)). To

maximize richness, V should be greater than or equal to NM . The middle

ground between min (M,N) and NM provides intermediate levels of richness

(see [12] for some simulated results). It should also be noted that smaller scale

effects, such as those due to mutual coupling, are not captured in this model

(see [13, 14]).


2.1.1.3 Double-bounce and keyhole MIMO analytical channel

models

In some cases, channel richness is compromised by the fact that some waves

follow common paths, thus limiting the independence between some signals;

as noted in [8, 15], this could be due either to a separation in free space

or to some sort of wave-guiding effect. A model that captures these effects

is the so-called double-bounce model, which is an extension of the single-

bounce model that includes a second ring of scatterers and is described by

the following equation:

HDB (N,M, V ) = ΦR(N,V1)Hi.i.d. (V1, V1)X (V1, V2)ΦT(M,V2)H. (2.8)

where V1 and V2 denote the number of scatterers in the first and second

ring, respectively. A special case of the model in (2.8), where the resulting

matrix has only a single nonzero eigenvalue, is the so-called keyhole or pinhole

channel [8, 15].

2.1.1.4 The Weichselberger MIMO analytical channel model

This model attempts to relax the separability between transmitter and re-

ceiver correlations by exploiting the eigenvalue decomposition of the corre-

sponding correlation matrices, shown below:

RT = UTΛTUHT

RR = URΛRUHR

, (2.9)

where UT , UR are unitary and ΛT , ΛR are diagonal matrices. The Weich-

selberger MIMO channel model is given by the following expression:

H = UR (Ω •Hi.i.d.)UHT , (2.10)

where Ω is a N × M coupling matrix that determines the average power

coupling between the transmit and receive eigenmodes and • denotes the

Schur-Hadamard product (element-wise multiplication). In fact, the Kro-

necker model is a special case of the Weichselberger model where the coupling

matrix Ω has rank 1. Other classes of random analytical MIMO channel mod-

els include propagation-based versions, such as:


• The finite scatterer model, which assumes a finite number of scatterers and

models the angles of departure and arrival, scattering coefficient and delay

for each scatterer [16]. This model allows incorporation of both single-

bounce and double-bounce scattering.

• The maximum entropy model [17], which attempts to incorporate proper-

ties of the propagation environment and system parameters via the maxi-

mum entropy principle so as to maximize the model’s match to the a priori

known information about the link.

• The virtual channel model which exploits the so-called “deconstructed”

MIMO channel representation proposed in [18], capturing the “inner”

propagation environment between virtual transmission and reception scat-

terers.

2.1.1.5 The Ricean MIMO channel model

Similarly to the case of scalar channels, when a line of sight (LOS) exists

between the transmitter and receiver, the channel is modeled as the sum of a

random part representing the non-LOS component and a deterministic part

that represents the LOS component. The well-known scalar Ricean channel

can be extended to the MIMO case as follows:

H =HR +

√KHD√

1 +K, (2.11)

where K ≥ 0 is the Rice factor (also called the “K factor”), HD denotes

the LOS deterministic channel matrix and HR denotes the random channel

matrix that can be modeled according to any of the MIMO channel models

presented above.

2.1.2 Physical channel models

In contrast to the analytical channel models, physical channel models focus on

the properties of the physical environment between transmitter and receiver

array. Two classes of physical channel model are briefly described below:

• Ray-tracing models

Ray-tracing (RT) models (see [6, 19]) are widely considered to be among

the most reliable deterministic channel models for wireless communica-


tions: they rely on the theory of geometrical optics to predict the way the

electromagnetic waves will reach the receiver after they interact with the

environment’s obstacles (which cause reflection, absorption, diffraction,

and so on). The Achilles’ heal of the RT approach is the need to know

in advance the physical obstacles of the propagation environment between

the transmitter and receiver. MIMO extensions of the RT approach have

been proposed in [20,21]. In the MIMO case, the pattern and polarization

of each antenna must be taken into account; however, this can be done in

a modular fashion, making the technique applicable to any known antenna

array configuration.

• Geometry-based stochastic physical models

Contrary to the deterministic nature of the RT approach described above,

which exploits the propagation environment’s geometry in a determin-

istic fashion, geometry-based stochastic physical models (GSCM) model

the scatterer locations in stochastic (random) terms, i.e. via their statis-

tical distributions. Beyond Lee’s original model of deterministic scatterer

locations on a circle around the mobile [22], various random scatterer dis-

tributions (including scatterer clustering) have since been proposed in,

for example, [23–26]. In the single-bounce approach each transmit/receive

path is broken into two sub-paths: transmitter-to-scatterer and scatterer-

to-receiver (described by their direction of departure, direction of arrival,

and path distance); the scatterer itself is modeled typically via the intro-

duction of a random phase shift. Multiple-bounce scattering has also been

proposed in order to address more complicated propagation environments

(see [27–29]). As mentioned above, MIMO versions of these models are

derived by considering the specific configuration and characteristics of the

antenna arrays on each side of the link.

2.1.3 Other extensions

The models mentioned above typically assume narrow-band propagation; in

other words, they are frequency-flat. Several wideband extensions have been

proposed in the literature to capture broadband communication links (see

e.g. references in [6]); these are especially relevant in view of the emerging

LTE/LTE Advanced and WiFi/WiMAX type systems, which typically use

OFDM and operate in bandwidths on the order of several tens of MHz. Also,

2.2 Single-user MIMO capacity 43

it was assumed that the propagation channel is time-invariant; time-varying

extensions have also been proposed [30, 31]; these are especially relevant in

cases of high user mobility. Mutual coupling between antenna elements (in

particular its role in affecting the channel’s spatial correlation properties) in

the above representative classes of MIMO channel models have been proposed

in [13, 14]. Finally, it should be mentioned that large collaborative efforts

have been undertaken over the last decade or so in order to propose MIMO

channel models that are fit for current and emerging wireless standards. These

include the COST259 [32], COST273 [29], IEEE 802.11n [33], Hiperlan2 [34],

Stanford University Interim (SUI) [26] and IEEE 802.16 [35]. The Spatial

Channel Model described in [36] is used as the basis for standards-related

simulations for 3GPP and 3GPP2.

2.2 Single-user MIMO capacity

In this section, expanding on the brief discussion on SU-MIMO capacity in

Chapter 1, we derive the capacity for the single-user MIMO channel. We

first derive the open-loop and closed-loop MIMO capacity for a fixed channel

realization, and then we study the performance of the capacity averaged over

random channel realizations.

2.2.1 Capacity for fixed channels

To begin with, we will focus on the case where the channel matrix H(t)

in (2.3) equals some fixed matrix H ∈ CN×M for all t, i.e., the channel

is time-invariant. We will further assume that H is known exactly to both

the transmitter and receiver. After dropping the time index t in (2.3) for

convenience, the input-output relationship of the channel reduces to

x = Hs+ n. (2.12)

The channel input s is subject to an average power constraint of P . The addi-

tive noise vector n has a circularly symmetric complex Gaussian distribution

with zero mean and covariance σ2IN .


By Shannon’s channel coding theorem [37], the capacity C(H, P/σ2

)of

the above channel, defined as the maximum data rate at which the decoding

error probability at the receiver can be made arbitrarily small with sufficiently

long codewords, is given by the maximum mutual information I(s;x) between

the input s and the output x, over all possible distributions for s that satisfy

the power constraint tr(E[ssH]) ≤ P . There is no loss of optimality here

in restricting s to have zero mean, since s− E [s] automatically satisfies the

power constraint if s does, and also yields the same mutual information with

x as s. Now,

I(s;x) = h(x)− h (x | s) (2.13)

= h(x)− h(n), (2.14)

where h(z) denotes the differential entropy of the random vector z, and

h (z | y) the conditional differential entropy of z given y.

The differential entropies can be evaluated using the following important

result about differential entropy (see, e.g., [38] for a proof): If z is any zero-

mean complex random vector with covariance E[zzH

]= Rz, then h(z) ≤

log |πeRz|, with equality holding if and only if z has a circularly symmetric

complex Gaussian distribution.

Thus in (2.14), we have h(n) = log∣∣πeσ2IN

∣∣. Maximizing I(s;x) therefore

amounts to maximizing h(x). Further, for any zero-mean s with covariance

E[ssH]= Rs, the channel output x is also zero-mean and has the covariance

σ2IN +HRsHH . Consequently,

h(x) ≤ log∣∣πe (σ2IN +HRsH

H)∣∣ , (2.15)

with equality if and only if x is circularly symmetric complex Gaussian. The

latter condition holds when the input s is itself circularly symmetric com-

plex Gaussian. We can therefore conclude that, among all zero-mean input

distributions with a given covariance Rs, the one that maximizes I(s;x) is

circularly symmetric complex Gaussian. Further, the corresponding mutual

information is

I(s;x) = log∣∣πe (σ2IN +HRsH

H)∣∣− log

∣∣πeσ2IN∣∣ (2.16)

= log

∣∣σ2IN +HRsHH∣∣

|σ2IN | (2.17)

= log∣∣IN +

(1/σ2

)HRsH

H∣∣ . (2.18)


Therefore the problem of determining the capacity C(H, P/σ2

)of the chan-

nel in (2.12) is reduced to that of finding the input covariance Rs that max-

imizes the RHS of (2.18), subject to the constraint tr (Rs) ≤ P :

C(H, P/σ2

)= max

Rs�0tr(Rs)≤P

log∣∣IN +

(1/σ2

)HRsH

H∣∣ . (2.19)

Clearly, in (2.19), any specific choice of the input covariance Rs satisfy-

ing tr (Rs) ≤ P will yield an achievable data rate, i.e., a lower bound on the

channel capacity. One such choice that is often of interest is Rs = (P/M) IM ,

which corresponds to an isotropic input, i.e., sending independent data

streams at the same power from each of the transmit antennas. The corre-

sponding achievable rate, which we will loosely term the “open-loop capacity”

of the channel and denote by COL(H, P/σ2

), is given by

COL(H, P/σ2

)= log

∣∣∣∣IN +P

Mσ2HHH

∣∣∣∣ . (2.20)

In order to motivate the choice of an isotropic input and the concept of

open-loop capacity, one can consider a situation where the transmitter has no

knowledge of the channel matrix H (but the receiver still knows it perfectly).

The isotropy of the additive noise n then suggests that the transmitter should

employ an isotropic input, hedging against its ignorance of the channel by

signaling with equal power in M orthogonal directions. More rigorous justi-

fications can be given for the optimality of an isotropic input in the context

of an ergodic channel model with spatially white noise [38].

2.2.1.1 Optimal input covariance

We will now sketch the derivation of the optimal input covariance Rs in

(2.19). The key idea here is to show that the MIMO channel can be decom-

posed into several single-input single-output (SISO) channels that operate

in parallel without interfering with each other, and must share the total

available transmit power of P . The optimal power allocation between these

SISO channels can then be obtained by a procedure commonly referred to

as “waterfilling” (the reason for the name will soon become clear). While

the derivation is of secondary importance for this book, we will provide this

rough proof because it reveals the important concept of spatial modes.


The decomposition of the MIMO channel into non-interfering SISO chan-

nels is based on the singular value decomposition (SVD) of the N×M channel

matrix H. This decomposition allows us to express H as

H = UΣVH , (2.21)

where U and V are N × N and M × M unitary matrices, respectively (so

UUH = UHU = IN , VVH = VHV = IM ) and Σ is an N × M diagonal

matrix. Each element of diag (Σ) is a singular value of H, i.e., the positive

square root of an eigenvalue of either HHH (if N ≤ M) or HHH (if N ≥ M).

Moreover, the columns of U are eigenvectors of HHH , and the columns of V

are eigenvectors of HHH.

Using (2.21) in (2.12), we get

x =(UΣVH

)s+ n ⇒ UHx =

(UHU

)Σ(VHs

)+UHn ⇒ x′ = Σs′ + n′,

(2.22)

where x′ = UHx, s′ = VHs, and n′ = UHn. Note that n′ has the same

distribution as n, since it is obtained by a unitary linear transformation of a

zero-mean circularly symmetric complex Gaussian vector whose covariance is

a multiple of the identity matrix. So the components of n′ are all independent,circularly symmetric, complex Gaussian random variables of mean 0 and

variance σ2. Note also that the signal terms of (2.22) are uncoupled, due to

the diagonal structure of Σ.

Let us assume now that rank(H) = r (where r ≤ min(M,N)). The matrix

Σ will then have r positive diagonal elements, which we will denote by λi, i =

1, . . . , r. These are the singular values of H, and λ2i , i = 1, . . . , r are the

eigenvalues of HHH . We will assume further that λ1 ≥ λ2 ≥ · · · ≥ λr. So

(2.22) can equivalently be written as:

x′i = λis

′i + n′

i, i = 1, . . . , r. (2.23)

(If r < N there are also N−r equations of the type x′i = n′

i, i = r+1, . . . , N ,

which contain no input signal information, and can therefore be neglected.)

Note that (2.23) describes an ensemble of r parallel, non-interfering SISO

channels, with gains λ1, λ2, . . . , λr and noise variance σ2. As a result, we can

depict the equivalent signal model as shown in Figure 2.2.

Assuming now that the transmitter allocates power Pi = E |s′i|2 to the ith

channel in (2.23), the SNR on the ith SISO channel is ρi = λ2iPi/σ

2, and

the rate achievable over it is Ri = log2 (1 + ρi). The overall rate achieved


MM MM

MM M

Fig. 2.2 Decomposition of the MIMO channel into r constituent SISO channels, where r

is the rank of H.

is∑

i Ri, i.e., the sum of the rates over the individual SISO channels. The

capacity of the MIMO channel is then obtained by maximizing the overall

rate over all power allocations, subject to the total transmit power constraint:

CCL(H, P/σ2) = maxP1,P2,...,Pr∑

i Pi=P

r∑i=1

log2

(1 +

Piλ2i

σ2

). (2.24)

The objective function in (2.24) is concave in the variables Pi and can be

maximized using Lagrangian methods (see [38]), yielding the following solu-

tion:

POpti =

(μ− σ2

λ2i

)+

,with μ chosen such thatr∑

i=1

POpti = P . (2.25)

Here (a)+= max (a, 0). The capacity of the channel is then given by [38]

CCL(H, P/σ2) =

r∑i=1

[log2

(λ2iμ

σ2

)]+. (2.26)

The covariance matrix of the transmitted signal s is given by:

Rs = V [diag (P1, · · · , PM )]VH . (2.27)


The optimal power allocation between the eigenmodes of the channel, given

in (2.25), can be computed by a procedure known as “waterfilling” [37]. Water

is poured in a two-dimensional container whose base consists of r steps, where

the “height” of each step is σ2/λ2j . If we let μ in (2.25) be the water level, the

optimal power allocated to the jth eigenmode POptj is the difference between

the water level and the height of the jth step (provided the water level is

higher than that step), where μ is set so the total allocated power is P .

Figure 2.3 illustrates the waterfilling algorithm for a MIMO channel with

three eigenmodes: λ1 > λ2 > λ3. If SNR is very low, the water level, as indi-

cated by the horizontal dotted line, covers only the first step. This indicates

that all the power P is allocated to the dominant eigenmode (POpt1 = P )

and no power is allocated to the others (POpt2 = POpt

3 = 0). As the SNR

increases, the other eigenmodes will be activated. For very high SNR, all

three modes are activated, and the difference in height between the steps is

insignificant compared to the water level. In this case, the power is allocated

approximately equally among the three eigenmodes: POpt1 ≈ POpt

2 ≈ POpt3 .

2.2.2 Performance gains

Having established the capacity of open- and closed-loop MIMO channels, we

now discuss the performance gains of MIMO relative to conventional single-

antenna techniques. In this section, we study in more detail the performance

gains mentioned briefly in Chapter 1, namely that the MIMO gains in the

low-SNR regime come about through antenna combining, and that the gains

in the high-SNR regime come from spatial multiplexing. We also consider

the capacity gains as the number of transmit and receive antennas increases

without bound.

Under a block-fading channel model, the channel realization is random

from block to block, and the capacity for each realization is a random variable.

A useful performance measure is the average capacity obtained by taking the

expectation of the capacity with respect to the distribution of H. The average

open-loop and closed-loop capacities are defined respectively as

COL(M,N,P/σ2) := EHCOL(H, P/σ2) (2.28)

CCL(M,N,P/σ2) := EHCCL(H, P/σ2). (2.29)


Low SNR Medium SNR High SNR

Transmit on a single eigenmode

Transmit on all eigenmodes with nearly

equal power

Transmit on multiple eigenmodes with different power

1 2 3Eigenmode index

0

Fig. 2.3 The waterfilling algorithm determines the optimal allocation of power among

parallel Gaussian channels that result from the decomposition of a MIMO channel. The

channel has three eigenmodes, and the power allocation is shown for low, medium, and

high values of SNR (P/σ2).

(In the context of fast fading channels, the average open-loop capacity as

defined here can also be interpreted as the “ergodic capacity” of the chan-

nel [39].) We will typically assume an i.i.d. Rayleigh distribution for the

components of H.

2.2.2.1 Low SNR

In the low-SNR regime where P approaches zero, the open-loop capacity for

a fixed channel H with rank r ≤ min(M,N) can be approximated as

COL(H, P/σ2) = log2 det

(IN +

P

Mσ2HHH

)(2.30)


= log2

r∏i=1

(1 +

P

Mσ2λ2i (H)

)(2.31)

≈ log2

(1 +

r∑i=1

P

Mσ2λ2i (H)

)(2.32)

= log2

[1 +

P

Mσ2tr(HHH

)](2.33)

≈ P

Mσ2tr(HHH

)log2 e, (2.34)

where λ2i (H) are the eigenvalues of HHH (and λi(H) are the singular values

of H). Equation (2.32) follows from the dominance of the linear terms for P

approaching zero, and (2.34) follows from the approximation

log2(1 + x) ≈ x log2 e (2.35)

for x approaching zero. Therefore the average open-loop capacity for i.i.d.

Rayleigh channels at low SNR is:

COL(M,N,P/σ2) ≈ P

Mσ2E[tr(HH)H

]log2 e (2.36)

=P

Mσ2E

[M∑

m=1

N∑n=1

|hn,m|2]log2 e (2.37)

= NP

σ2log2 e, (2.38)

where (2.38) follows from E

[∑Mm=1

∑Nn=1 |hn,m|2

]= MN for i.i.d. Rayleigh

channels. Hence at low SNR, the average open-loop capacity scales linearly

with the number of receive antennas N :

limP/σ2→0

C(OL)(M,N,P/σ2)

P/σ2= N log2 e. (2.39)

In the low-SNR regime, multiple transmit antennas do not improve the ca-

pacity, and the capacity of any (M,N) channel with M ≥ 1 is asymptotically

equivalent.

For the closed-loop capacity, waterfilling at asymptotically low SNR puts

all the power P into the single best eigenmode. (With i.i.d. Rayleigh channels,

the singular values will be unique with probability 1.) The average capacity

is therefore


C(CL)(M,N,P/σ2) = E

[max∑Pi≤P

r∑i=1

log2

(1 +

Pi

σ2λ2i (H)

)](2.40)

≈ E

[max∑Pi≤P

r∑i=1

Pi

σ2λ2i (H)

]log2 e (2.41)

≈ P

σ2E(λ2max(H)

)log2 e, (2.42)

where (2.41) follows from (2.35), and λ2max(H) is the maximum eigenvalue

value of HHH . Hence

limP/σ2→0

C(CL)(M,N,P/σ2)

P/σ2= E

(λ2max(H)

)log2 e. (2.43)

Because E(λ2max(H)

) ≥ max(M,N) for i.i.d. Rayleigh channels, closed-loop

MIMO capacity at low SNR benefits from combining at either the transmitter

or receiver.

2.2.2.2 High SNR

From (2.20), the average capacity for i.i.d. Rayleigh channels in the limit of

high SNR can be written as:

C(OL)(M,N,P/σ2) = E

⎡⎣min(M,N)∑

i=1

log2

(1 +

P

Mσ2λ2i (H)

)⎤⎦

≈ min(M,N) log2

(P

Mσ2

)+

min(M,N)∑i=1

E(log2 λ

2i (H)

), (2.44)

where (2.44) derives from the following approximation for large x:

log2(1 + x) ≈ log2(x). (2.45)

Because E(log2 λ

2i (H)

)> −∞ for all i, it follows that

limP/σ2→∞

COL(M,N,P/σ2)

log2 P/σ2

= min(M,N). (2.46)

Therefore open-loop MIMO achieves a multiplexing gain of min(M,N) at

high SNR.

For the closed-loop capacity at asymptotically high SNR, waterfilling puts

equal power in each of the min(M,N) eigenmodes. Therefore the average


capacity is

C(CL)(M,N,P/σ2) = E

⎡⎣ max∑

Pi≤P

min(M,N)∑i=1

log2

(1 +

Pi

σ2λ2i (H)

)⎤⎦

= E

⎡⎣min(M,N)∑

i=1

log2

(1 +

P/σ2

min(M,N)λ2i (H)

)⎤⎦

≈ min(M,N) log2

(P/σ2

min(M,N)

)+

min(M,N)∑i=1

E(log2 λ

2i (H)

), (2.47)

where (2.47) follows from (2.45). Hence

limP/σ2→∞

CCL(M,N,P/σ2)

log2 P/σ2

= min(M,N), (2.48)

and closed-loop MIMO achieves the same multiplexing gain as open-loop

MIMO (2.46) despite the advantage of CSIT. For both open- and closed-loop

MIMO, the multiplexing gain at high SNR requires multiple antennas at both

the transmitter and receiver.

2.2.2.3 Large number of antennas

When M and N go to infinity with the ratio M/N converging to α, and

the SNR remains fixed at P , the open-loop capacity per transmit antenna

converges almost surely to a constant [40]:

limM,N→∞M/N→α

COL(M,N,P/σ2)

M= log

[1 + S

(α,

P/σ2

α

)]+

1

P/σ2S

(α,

P/σ2

α

)

− 1

α+

1

αlog

[1 + P/σ2 − P/σ2

α+ S

(α,

P/σ2

α

)], (2.49)

where

S(α, ρ) =1

2

[ρ− ρα− 1 +

√(ρ− ρα− 1)

2+ 4ρ

].

Thus the capacity grows linearly with the number of antennas. The quantity

S(α, ρ) can be interpreted as the asymptotic SINR at the output of a linear

MMSE receiver for the signal from each of the M transmit antennas.

For the open-loop (M, 1) MISO channel with i.i.d. Rayleigh distribution,

the transmit power is distributed among the M antennas, and the average


capacity is given by

COL(M, 1, P/σ2) = E

[log2

(1 +

P

Mσ2Z,

)], (2.50)

where Z is a chi-square random variable with 2M degrees of freedom. Asymp-

totically, as M → ∞, the open-loop MISO capacity converges as a result of

the law of large numbers:

limM→∞

COL(M, 1, P/σ2) = log2(1 + P/σ2

). (2.51)

Therefore the result of transmit diversity (diversity is the only phenomenon

taking place in multi-antenna transmission with single-antenna reception) is

to remove the effect of fading when enough transmit antennas are available.

2.2.3 Performance comparisons

The CDF of the open-loop (4,1) MISO and (1,4) SIMO capacities are shown

in Figure 2.4 for i.i.d. Rayleigh channel realizations with SNR P/σ2 = 10.

The circles indicate the average capacities. For MISO channels, the average

capacity increases as M increases, and the CDF becomes steeper, indicating

there is less variation in the capacity as a result of diversity gain.

On the other hand, receiver combining for the SIMO channel results in

both diversity gain and combining gain. Increasing the number of receive

antennas results in a steeper CDF due to diversity and a shift with respect

to the open-loop MISO curve due to combining gains. The performance of

a (1, N) SIMO channel is equivalent to that of a (N, 1) closed-loop MISO

channel.

Figure 2.5 shows the average capacity of various link configurations versus

SNR. The (4, 1) OL MISO capacity yields a small improvement over the SISO

performance. The (4, 1) CL, and (1, 4) performance is better as a result of

coherent transmitter or combining gain, but the slope of the capacity curve

with respect to log2 P/σ2 is the same as SISO’s. At high SNR, the open-loop

and closed-loop MIMO techniques achieve a multiplexing gain of 4, indicated

by the slope of the capacity. For asymptotically high SNR, the open-loop and

closed-loop capacities are equivalent, and there is already negligible difference

for SNRs greater than 20 dB. For MIMO, every doubling (3 dB increase) in


SNR results in 4 bps/Hz of additional capacity. For SIMO or MISO, every

doubling results in only 1 bps/Hz of additional capacity.

Figure 2.6 shows the average capacity of the same link configurations for

a lower range of SNRs. At very low SNRs, the optimal transmission strategy

benefits from diversity and combining but not from multiplexing. Compared

to (1, 4) SIMO, additional transmit antennas under (4, 4) OL MIMO do not

provide any benefit. Using knowledge of the channel at the transmitter, (4, 4)

CL MIMO achieves additional capacity by steering power in the direction of

the channel’s dominant eigenmode. To better visualize the relative gains due

to multiple antennas compared to SISO, Figure 2.7 shows the ratios of the

MIMO, SIMO, and MISO average capacities versus the SISO average capacity

as a function of SNR. For (4, 4) OL MIMO, the ratio is 4 for both low and

high SNRs but dips below 4 in between.

For CL MIMO, the number of transmitted streams as determined by wa-

terfilling depends on the SNR. Figure 2.8 shows the average number of trans-

mitted streams (average number of eigenmodes with nonzero power) for dif-

ferent antenna configurations as a function of SNR. For SNRs below -15 dB,

capacity is achieved by transmitting a single stream for all cases. As the SNR

increases, the probability of transmitting multiple streams increases. For (2,2)

and (4,4) multiplexing the maximum number of streams min(M,N) occurs

with probability 1 for SNRs of at least 30 dB. For (2,4), full multiplexing

with probability 1 occurs for SNRs of at least 10 dB.

Figure 2.9 shows the average capacity versus the number of antennas M

for (M,M) MIMO and (1,M) SIMO. Because the MIMO capacity is roughly

M log2(P/σ2) for high SNR, the slope of the curve versus M depends on the

SNR.

2.3 Transceiver techniques

The previous section describes the theoretical capacity of MIMO links but

only hints at the transceiver (transmitter and receiver) structure required to

achieve those rates. In this section we discuss transceiver implementation for

achieving open- and closed-loop capacity and a number of relevant suboptimal

techniques, including linear receivers and space-time coding.

2.3 Transceiver techniques 55

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

CD

F

(1,4)(1,1) (4,1)

Fig. 2.4 CDF of capacity for SISO, open-loop MISO and SIMO channels for i.i.d. Rayleigh

channels. The MISO channel increases reliability by providing diversity gain. The SIMO

channel provides both diversity and combining gain.

2.3.1 Linear receivers

Let us consider the received signal (2.3) for the (M,N) MIMO channel where

the data stream from the mth transmit antenna is highlighted:

x = hmsm +∑j =m

hjsj + n, (2.52)

where hm is the mth column of the channel matrix H. We assume that the

power of the mth stream is Pm := E

[|sm|2

]and that the noise vector n is

ZMSW Gaussian with covariance σ2IM .

We are interested in the class of linear receivers which computes a decision

statistic rm for the mth data stream by correlating the received signal x with

an appropriately chosen vector wm:

rm = wHmx (2.53)

= (wHmhm)sm +

∑j =m

(wHmhj)sj +wH

mn. (2.54)


0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

SNR (dB)

Ave

rage

cap

acity

(bps

/Hz)

(4,4) CL(4,4) OL(4,1) CL, (1,4) OL/CL(4,1) OL(1,1)

Fig. 2.5 Average capacity versus SNR for i.i.d. Rayleigh channels. At high SNR, (4,4)

OL and CL MIMO provide a multiplexing gain of 4.

-20 -15 -10 -5 00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR (dB)

Ave

rage

cap

acity

(bps

/Hz)

(4,4) CL(4,4) OL(4,1) CL, (1,4) OL/CL(4,1) OL(1,1)

Fig. 2.6 Average capacity versus SNR for i.i.d. Rayleigh channels. At low SNR, (4,4)OL

MIMO, (4,1)CL MISO, and (1,4)OL/CL SIMO provide a power gain of 4 as a result of

combining.


-20 -10 0 10 20 30 400

1

2

3

4

5

6

7

8

9

10

SNR (dB)

MIM

O g

ain

(4,4) CL

(4,4) OL

(4,1) CL, (1,4) OL/CL,

(4,1)OL

Fig. 2.7 Ratio of MIMO, SIMO, and MISO average rates versus the SISO average rate

as a function of SNR for i.i.d. Rayleigh channels.

-20 -10 0 10 20 30 401

1.5

2

2.5

3

3.5

4

SNR (dB)

Ave

rage

num

ber o

f tra

nsm

itted

stre

ams

(4,4)

(2,2)

(2,4)

Fig. 2.8 Average number of transmitted streams for CL MIMO with i.i.d. Rayleigh chan-

nels. For lower SNRs, only a single stream is transmitted. Spatial multiplexing occurs at

higher SNRs.


SNR = 10dB

SNR = 30dB

1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

, number of antennas

Ave

rage

cap

acity

(bps

/Hz)

Fig. 2.9 Average capacity versus number of antennas M for i.i.d. Rayleigh channels. The

slope of the MIMO capacity depends on the SNR.

The output signal-to-noise ratio (SNR), defined as the ratio of the receiver

output power of the desired stream to the receiver output power of the ther-

mal noise, is given by

E[|wH

mhmsm|2]E [|wH

mn]=

|wHmhm|2Pm

||wm||2σ2. (2.55)

In contrast to the SNR of the received signal P/σ2 defined in Section 2.1,

we emphasize that the output SNR (2.55) is defined at the output of the

receiver processing. The output signal-to-interference-plus-noise ratio (SINR)

is defined as the ratio of the receiver output power of the desired stream to

the sum of the receiver output power of the thermal noise and interference

from the other streams. Because the thermal noise and data streams are

uncorrelated, the SINR is given by:

E

[∣∣wHmhmsm

∣∣2]E

[∣∣∣wHm

(∑j =m hjsj + n

)∣∣∣2] =

∣∣wHmhm

∣∣2 Pm∑j =m |wH

mhj |2 Pj + ‖wm‖2 σ2. (2.56)


We consider two linear receivers: the matched filter (MF) receiver and the

minimum mean-squared error (MMSE) receiver. The MF is defined as the

correlator matched to the desired stream’s channel:

wm = hm. (2.57)

Because the noise vector is Gaussian, this receiver maximizes the output

SNR [41], but it is oblivious to the interference from the other data streams.

It requires knowledge of the desired stream’s channel but no knowledge of

the other streams’. From (2.55), the output SNR is ‖hm‖2 Pm/σ2, and the

output SINR is

ΓMF,m =‖hm‖4 Pm

‖hm‖2 σ2 +∑

j =m |h∗mhj |2 Pj

. (2.58)

The MF receiver is also known as the maximal ratio combiner (MRC) because

it weights and combines the received signal components to maximize the

output SNR.

The MMSE receiver is a more sophisticated linear receiver that accounts

for the presence of interference by minimizing the mean-squared error be-

tween the receiver output and the desired data stream sm:

wm = argminw

E

[∣∣wHx− sm∣∣2]

= argminw

wH(HPHH + σ2IN

)w − 2wHhmPm + Pm

=(HPHH + σ2IN

)−1

hmPm,

where P := diag(P1, . . . , PM ) is the diagonal matrix of powers. Using the

matrix inversion lemma [42] for invertible A:

(A+ bbH

)−1

= A−1 − A−1bbHA−1

1 + +bHA−1b, (2.59)

and defining X :=∑

j =m hjhHj Pj + σ2IN , we can write the MMSE receiver

as

wm =(HPHH + σ2IN

)−1

hmPm

=(X+ hmhH

mPm

)−1hmPm

= X−1hmPm − X−1hmhHmX−1hmP 2

m

1 + hHmX−1hmPm


=X−1hmPm

1 + hHmX−1hmPm

. (2.60)

Using (2.56) and (2.60), the SINR of the mth stream at the MMSE receiver

output is

ΓMMSE,m =wH

mhmPmhHmwm

wHmXwm

=Pm

σ2hHm

⎛⎝IN +

∑j =m

Pj

σ2hjh

Hj

⎞⎠−1

hm. (2.61)

The MMSE receiver is the linear receiver which maximizes the SINR [43],

and in this sense, it is often said to be the optimal linear receiver. We note

that the MMSE receiver for a particular data stream can also be obtained

by whitening the total noise plus interference affecting that stream, and then

computing the matched filter for the equivalent channel after whitening.

As given in (2.60), the MMSE receiver requires knowledge of all chan-

nels. This requirement is reasonable for many situations where pilot signals

are transmitted from each of the antennas. (See Section 2.7 for a discussion

on acquiring channel estimates.) If the channel estimates are unreliable or

unavailable, blind receivers techniques could be used [44].

Now suppose that M and N both go to infinity and M/N → α. In this

large-system limit, it can be shown that [43]

ΓMF,m → P/σ2

α(1 + P/σ2)(2.62)

and

ΓMMSE,m → 1

2

⎡⎣( P

σ2α− P

σ2− 1

)+

√(P

σ2α− P

σ2− 1

)2

+4P

σ2α

⎤⎦ . (2.63)

Figure 2.10 shows the mean SINR (averaged over transmit antennas as well

as i.i.d. Rayleigh channel realizations) versus average SNR P/σ2 for both

the MF and MMSE linear receivers. In each case, the solid lines show the

results for M = 4 and N = 4 (as in (2.58) and (2.61)), while the dashed

lines show the asymptotic results for α = 1 (as in (2.62) and (2.63)). It can

be observed that, in contrast to the interference-aware MMSE receiver, the

SINR attainable with the interference-oblivious MF receiver saturates as the

SNR is increased, indicating that it is interference-limited. (In fact, it can be


0 5 10 15 20-5

0

5

10

15

SNR (dB)

Mea

n SI

NR

(dB)

MF, asymptotic

MF,

MMSE, asymptotic

MMSE,

Fig. 2.10 Mean SINR per transmit antenna for MF and MMSE receivers. The asymptotic

results assume that M and N both go to infinity with M/N → 1.

shown [39] that the linear MMSE receiver attains the optimal multiplexing

gain of min(M,N), i.e., the rate achievable with it as a function of SNR

exhibits the same slope at high SNR as the capacity of the MIMO channel.)

The trends exhibited by the asymptotic cases are already apparent for a link

with relatively few antennas.

2.3.2 MMSE-SIC

The performance of the MMSE receiver could be improved by following it

with a nonlinear successive interference cancellation (SIC) stage, shown in

Figure 2.11. Suppose that we detect the data symbol s1 from the first antenna.

Its SINR is ΓMMSE,1, given by (2.61). Assuming s1 is detected correctly and

assuming the receiver has ideal knowledge of h1, it can be cancelled from the

received signal x, yielding:

x− h1s1 =

M∑j=2

hjsj + n. (2.64)


MMSE, symbol 1w.r.t. interf. 2, 3

Cancel symbol 1 MMSE, symbol 2w.r.t. interf 3

Cancel symbol 2 MF, symbol 3

Fig. 2.11 MMSE-SIC detector for M = 3 symbols. Symbols are detected and cancelled

in order, yielding estimates s1, s2, s3.

Given (2.64), data symbol s2 can be detected using an MMSE receiver. Be-

cause there is no contribution from s1 in (2.64), its SINR is:

Γ2 =P2

σ2hH2

⎛⎝ M∑

j=3

IN +Pj

σ2hjh

Hj

⎞⎠−1

h2.

If we successively detect the data symbols in order s3, s4, . . . ,M and cancel

their contributions, the mth stream (m = 1, . . . ,M − 1) experiences inter-

ference from streams m+ 1,m+ 2, . . . ,M . The SINR for the mth stream is

therefore

Γm =Pm

σ2hHm

⎛⎝ M∑

j=m+1

IN +Pj

σ2hjh

Hj

⎞⎠−1

hm. (2.65)

The Mth stream is detected in the presence of only Gaussian noise. Using a

matched filter, its SNR is

ΓM =PM

σ2‖hM‖2 . (2.66)

In the discussion of the MMSE-SIC detector (and the MF and MMSE

detectors), we have focused on the detection of the data symbols sm without

any regard to channel encoding. In general, these data symbols are the output

of a channel encoder, and we show in the next section how the MMSE-SIC

detector in conjunction with a channel decoder can be used to achieve the

open-loop MIMO capacity.


2.3.3 V-BLAST

The open-loop capacity for a fixed (M,N) MIMO link can be achieved using

the vertical BLAST (V-BLAST) architecture, which uses an MMSE-SIC re-

ceiver structure where the interference cancellation is performed with respect

to the decoded data streams. In the MIMO literature, the term “V-BLAST”

is used to describe a variety of transceiver architectures and can be applied to

block- or fast-fading channels. However, we use the term to refer specifically

to the transmitter architecture shown in Figure 2.12 where the information

data stream is multiplexed into M lower-rate streams that are independently

encoded. This transmit architecture is sometimes referred to as per-antenna

rate control (PARC) because the rate of each antenna’s data stream is ad-

justed based on the channel realization H.

From (2.3), the encoded transmitted signal vector for symbol time t is

denoted as s(t), and we let{s(t)}

denote the stream of vectors associated

with a coding block. Similarly we let{x(t)}denote the corresponding block of

received signals. We assume that the channel is stationary for the duration of

the coding block and that each stream has power Pm = P/M , m = 1, . . . ,M .

Applying an MMSE detector to the received signal vectors{x(t)}, the output

SINR is (2.61):

Γ1 =P

Mσ2hH1

⎛⎝IN +

M∑j=2

P

Mσ2hjh

Hj

⎞⎠−1

h1. (2.67)

If the data stream for the first antenna{s(t)1

}is encoded using a capacity-

achieving code corresponding to rate log2(1 + Γ1), then it can be decoded

without error. Using ideal knowledge of h1, its contribution to the received

signal{x(t)}can be cancelled. In general, if the mth data stream is encoded

with rate log2(1 + Γm), where from (2.65),

Γm =P

Mσ2hHm

⎛⎝IN +

M∑j=m+1

P

Mσ2hjh

Hj

⎞⎠−1

hm, (2.68)

then it can be decoded and cancelled from the received signal so that data

streams m + 1,m + 2, . . . ,M do not experience interference from it. Using

(2.68) and the matrix identities [42]


det(A)

det(B)= det(B−1A) and

det(I+AB) = det(I+BA),

the rate achievable by stream m can be written as

log2(1 + Γm) = log2

⎡⎢⎣1 + P

Mσ2hHm

⎛⎝IN +

M∑j=m+1

P

Mσ2hjh

Hj

⎞⎠−1

hm

⎤⎥⎦

= log2 det

⎡⎢⎣IN +

P

Mσ2

⎛⎝IN +

M∑j=m+1

P

Mσ2hjh

Hj

⎞⎠−1

hmhHm

⎤⎥⎦

= log2

det(IN +

∑Mj=m

PMσ2hjh

Hj

)det(IN +

∑Mj=m+1

PMσ2hjhH

j

) . (2.69)

If we add the achievable rates for all M streams, then from (2.69), all the

terms are cancelled except for the numerator of the rate for stream 1. The

achievable sum rate is therefore

M∑m=1

log2(1 + Γm) = log2 det

⎛⎝IN +

M∑j=1

P

Mσ2hjh

Hj

⎞⎠

= log2 det

(IN +

P

Mσ2HHH

). (2.70)

Noting the equivalence between (2.70) and (2.20), we conclude that the PARC

strategy with an MMSE-SIC receiver achieves the open-loop MIMO capacity

for block-fading channels.

For a fast-fading channel with i.i.d. Rayleigh distribution, using the statis-

tics of H, we can set the rate of stream m to be

Rm = EH [log2 (1 + Γm)] , (2.71)

where Γm is from (2.68). Then, from (2.70), the achievable sum rate is

M∑m=1

[EH log2(1 + Γm)] = EH

[log2 det

(IN +

P

Mσ2HHH

)]. (2.72)

Therefore the V-BLAST architecture also achieves the ergodic capacity for

fast-fading channels. We emphasize that for a block-fading channel, the rates


Stream 1: Coding,

modulation

Stream 2: Coding,

modulation

Stream 3: Coding,

modulation

MMSE-SIC

De-mux

MMSE, stream 1w.r.t. interf. 2, 3

Cancel stream 1 MMSE, stream 2w.r.t. interf 3

Cancel stream 2 MF, stream 3

Mux

decode, stream 3

decode, stream 2

decode, stream 1

Fig. 2.12 Transceiver for achieving OL-MIMO capacity using the V-BLAST transmit

architecture and an MMSE-SIC receiver.

are set as a function of the realization H, while for a fast-fading channel, the

rates are based on the statistics of H.

Even though we have assumed that the data streams are decoded and

cancelled in order from m = 1 to M , we note that the sum rate computed via

(2.70) is independent of the order. Therefore, the OL MIMO capacity can be

achieved for any ordering, as long as each antenna’s stream is encoded with

the appropriate rate as determined by the particular ordering.

The optimality of V-BLAST and PARC with MMSE-SIC was shown in [45]

and is based on the sum-rate optimality of the MMSE-SIC for the multiple

access channel [46]. This topic will be revisited in Chapter 3. While the

PARC strategy assumes equal power on each stream, the throughput can be

increased by optimizing the power distribution among the streams [45] using

knowledge of H at the transmitter.


2.3.4 D-BLAST

In contrast to V-BLAST, where data streams for each antenna are encoded

independently, Diagonal BLAST (D-BLAST) is an alternative technique for

achieving the open-loop capacity that transmits the symbols for each coding

block from all M antennas. Figure 2.13 shows the D-BLAST transceiver

architecture. The information stream is encoded as U blocks of ML symbols.

In the context of D-BLAST, each coding block is known as a layer. Layer

u = 1, . . . , U consists of two subblocks of L symbols:{b(u)1

}and

{b(u)2

}. The

blocks are transmitted in a staggered fashion so that L symbols{b(u)2

}are

transmitted from antenna 2, followed by L symbols{b(u)1

}transmitted from

antenna 1. The layer u transmission on antenna 1 occurs at the same time as

the layer u+1 transmission on antenna 2. During the first L symbol periods,

symbols{b(1)2

}are transmitted from antenna 2, and nothing is transmitted

from antenna 1. During the last L symbol periods,{b(U)1

}are transmitted

from antenna 1, and nothing is transmitted from antenna 2.

MLayer u: Coding,

modulation

Mux andModulo-M

shift

h

h

time

MMSE-SIC

De-mux

Fig. 2.13 The D-BLAST transmitter cyclicly shifts the association of each stream with

all M antennas. The modulo-M shift is illustrated for M = 2 antennas and U layers.

To decode layer 1, symbols{b(1)2

}are detected using a matched filter in

the presence of thermal noise. The output SNR is


Γ2 =P

2σ2|h2|2. (2.73)

The symbols{b(1)1

}are detected using an MMSE receiver in the presence of

interference from antenna 2 from symbols{b(2)2

}. The output SINR is

Γ1 =P

2σ2hH1

(IN +

P

2σ2h2h

H2

)−1

h1. (2.74)

If the 2L symbols of layer 1 are encoded using a compound code at a rate

R < log2(1 + Γ1) + log2(1 + Γ2), then these symbols can be reliably decoded

and canceled from the received signal stream.

The decoding of layer u follows the same procedure. Symbols{b(u)2

}are

detected using a matched filter because symbols from the previous layer trans-

mitted on antenna 1 have been cancelled. Then symbols{b(u)1

}are detected

using an MMSE in the presence of interference from{b(u+1)2

}. If U layers

are transmitted, the achievable rate is

U

U + 1[log2(1 + Γ1) + log2(1 + Γ2)] , (2.75)

where the fraction is due to the empty frames during the first and last layers.

This overhead vanishes as U increases.

The procedure described for the M = 2 antenna case can be generalized

for more antennas, so that the symbols for a single layer are staggered over

ML symbol periods and transmitted over all M antennas. The symbols from

antenna m = 1, ...,M are detected in the presence of interference from anten-

nas m+1, . . . ,M . The SINR achieved for detecting symbols from antenna m

is (2.68), and the overall achievable rate (for large U) is the open-loop MIMO

capacity (2.70).

In practice, because the transmitter does not have knowledge of the chan-

nel, it does not know at what rate to encode the information. The receiver

could estimate the channel H, determine the channel capacity, and feed

back this information to the transmitter. The D-BLAST encoder needs to

know only the MIMO capacity whereas the V-BLAST encoder needs to know

the achievable rates of each stream. D-BLAST would therefore require less

feedback. However, due to the difficulty in implementing efficient compound

codes, the V-BLAST architecture is more commonly implemented.


2.3.5 Closed-loop MIMO

If the channel H is known at the transmitter, one can achieve capacity by

transmitting on the eigenmodes of the channel, as discussed in Section 2.2.1.1.

The corresponding transceiver structure is shown in Figure 2.14. The infor-

mation bit stream is first multiplexed into min(M,N) lower-rate streams,

and the streams are encoded independently according to the rates deter-

mined by waterfilling. Given the SVD of the MIMO channel H = UΛVH ,

the min(M,N) streams are precoded using the first min(M,N) columns of

the M × M unitary matrix V. At the receiver, the N × N linear transfor-

mation UH is applied, and the elements of the first min(M,N) rows are

demodulated and decoded. The information bits for the min(M,N) streams

are demultiplexed to create an estimate for the original bit stream.

For the special case of the (M, 1) MISO channel, the data stream is pre-

coded with the unit-normalized complex conjugate of the channel vector

h ∈ C1×M : (v = hH/ ‖h‖). This weighting is sometimes known as maxi-

mal ratio transmission (MRT), and it is the dual of MRC receiver for the

(1, N) SIMO channel.

MM M

Stream 1: Coding,

modulation

Stream : Coding,

modulation

Stream : Coding,

modulation

MMux

Stream 1: Coding,

modulationDe-mux

Fig. 2.14 Capacity-achieving transceiver for CL MIMO based on the SVD of H. The

power allocated to each stream is determined by waterfilling.

2.3.6 Space-time coding

If the channel is known at the transmitter, the SVD-based strategy described

in Section 2.3.5 achieves the closed-loop capacity for any (M,N) link. If the

channel is not known at the transmitter, the strategies for achieving open-


loop capacity described in Section 2.3.5 apply only when M ≤ N . Space-time

coding is a class of techniques for achieving diversity gains in MISO channels

when the channel state information is not known at the transmitter [47] [48].

Multiple receive antennas could be used to achieve combining and additional

diversity gains. We will outline the basic principles of space-time block codes

(STBCs), which are illustrative and representative of what space-time coding

can achieve in MISO channels. The space-time block-coding transmission

architecture is shown in Figure 2.15.

M

Space-time block codingCoding,

modulation

Fig. 2.15 In a space-time block-coding transmission architecture, the coded symbols{b(t)

}are mapped to the transmitted symbol vector using, for example, (7.2)

.

In this architecture, a data stream is encoded using an outer channel en-

coder, and a space-time block encoder maps a block of Q encoded symbols

b1, . . . , bQ onto theM antennas over L symbol periods. This mapping is repre-

sented by an M×L matrix S, where the (m, l)th element of S (m = 1, . . . ,M ,

l = 1, . . . , L) is the symbol transmitted from antenna m during symbol pe-

riod l. In general, each element of S is a linear combination of b1, . . . , bQ

and of the respective complex conjugates of these symbols b∗1, . . . , b∗Q. The

parameters L and R := Q/L are known respectively as the code delay and

code rate of space-time block code S. Typically, the mapping parameters are

chosen so that L ≥ M and R ≤ 1. The performance of a space-time code

can be measured by its diversity order which we define as the magnitude of

the slope of the average symbol error rate at the receiver versus SNR (in a

log-log scale).

For an (M, 1) channel, the optimal (maximum) diversity order is M and

can be achieved if SSH is proportional to the identity matrix IM [49]. It is

also desirable for a code to be full rate (i.e., R = 1 and Q = L) and delay

optimal (i.e., L = M) so that the code is time efficient. A space-time block


code is ideal if it is full rate and delay optimal (so that L = M = Q) and if

it achieves maximum diversity order.

For the case of M = 2 transmit antennas, a very popular and remarkably

efficient space-time block code (the one that really defined the class of space-

time block codes) is the Alamouti space-time block code [50]. Given a sequence

of encoded symbols{b(t)}(t = 0, 1, . . .), each pair of symbols on successive

time intervals b(2j) and b(2j+1) (j = 0, 1, . . .) is transmitted over the two

antennas on intervals 2j and 2j + 1 as follows:

s(2j) =

[b(2j)

b(2j+1)

]and s(2j+1) =

[−b∗(2j+1)

b∗(2j)

].

This code is ideal because it achieves maximum diversity order M = 2 with

L = M = Q = 2. Moreover, it quite remarkably achieves the open-loop ca-

pacity for the (2,1) MISO channel for any SNR if an outer capacity-achieving

scalar code is used [51] [52]. It also achieves the optimal diversity/ multiplex-

ing tradeoff (see e.g. [53]). For a (2, N) channel with N > 1, the Alamouti

STBC with maximal ratio combining in general does not achieve the capac-

ity. Not surprisingly, due to its remarkable properties, the Alamouti code has

been used in several wireless standards.

To date, no ideal space-time block codes have been found for M > 2.

However, quasi-orthogonal STBCs have also been proposed that approach

the open-loop capacity in the (4,1) case [54] [55]. While generalizations of the

quasi-orthogonal concept (and other space-time coding techniques) to arbi-

trary numbers of antennas have been suggested [56], it should be emphasized

that the marginal diversity gains for open-loop MISO techniques diminish

as the number of antennas increases. This fact, coupled with the additional

overhead required to the channels, reduces the incentive for using too many

(M > 4) antennas for space-time coding.

Besides STBCs, other classes of space-time codes include space-time trellis

codes [47], linear dispersion codes [57], layered turbo codes [58], and lattice

space-time codes [59].

2.3.7 Codebook precoding

If CSIT is ideally known, precoding with waterfilling achieves the closed-loop

MIMO capacity. In many practical cases, it is not possible to obtain reliable


CSIT (see Section 4.4). Isotropic transmission is suboptimal, and as we saw

in Section 2.2.3, the performance gap between OL-MIMO and CL-MIMO is

significant at lower SNRs.

Another suboptimal alternative is to use precoding matrices that are cho-

sen from a finite discrete set known as a codebook. (The precoding matrices are

sometimes known as codewords, but they are not to be confused with the code-

words associated with channel encoding.) The codebook is known by both the

transmitter and receiver. Under codebook precoding, typically the receiver

estimates the channel H and sends information back to the transmitter to

indicate its preferred codeword. Using B feedback bits, the receiver can index

up to 2B codewords in the codebook. A block diagram is shown in Figure

2.16, where the codebook B consists of 2B precoding matrices G1, . . . ,G2B .

This precoding technique is sometimes known as limited feedback precoding.

bits

Fig. 2.16 Block diagram for codebook precoding. The user estimates the channel H and

feeds back B bits to indicate its perfered precoding vector. The codebook B consists of 2B

codeword matrices and is known by the transmitter and the user.

In general, the transmitter can send up to min(M,N) data streams. If we

let J ≤ min(M,N) denote the number of streams, and u ∈ CJ be the vector

of data symbols, then the precoding matrix G ∈ B is size M × J , and the

transmitted signal is given by s = Gu. From 2.3, the received signal is

x = HGu+ n. (2.76)

When J = 1 and the input covariance has rank 1, precoding is often known

as beamforming.

If the MIMO channel is changing sufficiently slowly, the mobile feedback

could be aggregated over multiple feedback intervals so that the aggregated

bits index a larger codebook. In general, a larger codebook implies more

accurate knowledge of the MIMO channel at the transmitter, resulting in im-


proved throughput. By aggregating the feedback bits over multiple intervals,

the codewords can be arranged in a hierarchical tree structure so that the

feedback on a given interval is an index of codewords that are the “children

nodes” of a codeword indexed by previous feedback [60]. Temporal correlation

of the channel can also be exploited by adapting codebooks over time [61] or

by tracking the eigenmodes of the channel [62] [63] [64].

2.3.7.1 Single-antenna receiver, N = 1

Let us consider the problem of designing a codebook B for the case of a

single-antenna receiver N = 1. In this case, the codebook consists of 2B

beamforming vectors: B = {g1, . . . ,g2B}, with gb ∈ CM×1. Assuming that

the channel h ∈ C1×M can be estimated ideally, the user chooses the code-

word in B which maximizes its rate:

maxg∈B

log2

(1 + |hg|2 P

σ2

)= argmax

g∈B|hg|. (2.77)

If the channel realizations are drawn from a finite, discrete distribution of 2B

M -dimensional vectors, one would design the codebook to consist of these

vectors. The rate-maximizing codeword would be the (normalized) channel

vector which corresponds to the maximal ratio transmitter (MRT). Assuming

the channels could be estimated without error and the B feedback bits from

each user could be received without error, the transmitter would achieve

ideal CSIT. In practice, because the channel realizations are drawn from

a continuous distribution, the codewords should be designed to optimally

span the distribution, as determined by the channel correlation and desired

performance metric.

At one extreme, the antennas are spatially uncorrelated, and the MISO

channel coefficients each have an i.i.d. Rayleigh distribution. In this case the

normalized realization h/||h|| is distributed uniformly on an M -dimensional

unit hypersphere. The optimal rate maximizing strategy is to distribute the

2B codewords as uniformly as possible on the surface of the hypersphere [65].

This problem is known as the Grassmannian line packing problem: design the

codebook B to maximize the minimum distance between any two codewords√1−max

i =j|gH

i gj |2. (2.78)


At the other extreme, the antennas are totally correlated, for example in a

line-of-sight channel with zero angle spread. Figure 2.17 shows a linear array

with M elements lying on the x-axis with uniform spacing d and a user with

direction θ with respect to the x-axis. Let us consider the channel response h

measured by a user lying in the general direction θ ∈ [0◦, 360◦). If the channelcoefficient of the first element is h1 = α exp(jγ), then the coefficient at the

mth element(m = 1, ...,M) is

hm(θ) = α exp

(2πjd

λ(m− 1) cos θ + jγ

), (2.79)

where λ is the carrier wavelength. We can use MRT to create a beamforming

vector g(θ∗) in the direction θ∗ by matching the phase of the beamforming

weight gm(θ∗) to the phase of the channel coefficient hm(θ∗), modulo the

phase offset γ. With d = λ/2, the resulting MRT beamforming vector is

g(θ∗) =1√M

⎡⎢⎢⎢⎢⎣

1

exp (πj cos θ∗)...

exp (πj(M − 1) cos θ∗)

⎤⎥⎥⎥⎥⎦ . (2.80)

Using this beamforming vector, the SNR of a user lying in the direction θ is

|hH(θ)g(θ∗)|2P/σ2. (2.81)

The MRT beamforming vector creates a directional beam (pointing in the

direction θ∗) in the sense that the transmitted signal is co-phased to maximize

the SNR of a user lying direction θ = θ∗. Figure 2.18 shows the MRT beam

response for a linear array with M = 4 elements and a desired direction

θ∗ = 105◦. (The elements themselves are directional and pointing in the

direction θ = 90◦, as described in Section 6.4.3. Otherwise, there would also

be a response peak in the direction θ∗ +180◦.) Codewords could be designed

to form directional beams spanning a desired range. For example, if users

lie in a 120-degree sector θ ∈ [30◦, 150◦], we could choose to span this range

using four MRT beams with directions {45◦, 75◦, 105◦, 135◦}. A user could

determine its best codeword from (2.77) and indicate its preference with only

B = 2 bits.

More general design techniques known as robust minimum variance beam-

forming can be used to design beamforming vectors for arbitrary antenna

array configurations that are robust enough to withstand mismatch between


Element index

θ

…

User

……

dθ∗

Fig. 2.17 An M -element linear array with inter-element spacing d. The direction of the

user is θ, and a beam is pointed in the direction θ∗ = 105◦.

0 50 100 150 200 250 350-50

-40

-30

-20

-10

0

(degrees)

Fig. 2.18 The directional response as a function of the user direction θ for a MRT beam-

forming vector (2.80) pointing in the direction θ∗ = 105◦.


measured and actual channel state information h(θ) [66]. The design of equal-

gain beamformers with limited feedback [67] is also relevant for antenna ar-

rays where the amplitudes of the channel coefficients are highly correlated.

For intermediate situations where the spatial channels are neither fully

correlated nor totally uncorrelated, systematic codebook designs have been

proposed in [68]. Codebooks can also be designed implicitly using a train-

ing sequence of channel realizations drawn from a given spatial correlation

function. This technique, based on the Lloyd-Max algorithm [69], is effective

for creating specifically tailored codebooks for arbitrary spatial correlations.

The training sequence {Hj}NTSj=1 is of size NTS , and its elements Hj are real-

izations of MIMO channels drawn from a given spatial correlation function.

If we let μ(Hj ,gi) be a performance metric for a given channel realization

and codebook vector, the algorithm iteratively maximizes the average per-

formance metric

maxB

1

NTS

2B∑i=1

∑Hj∈Ri

μ(Hj ,gi), (2.82)

where Ri is the partitioned region of the training sequence associated with

codeword gi. The size of the training sequence needs to scale at least linearly

with the number of desired codewords to achieve good performance [69],

hence the complexity of codebook design scales at least exponentially with

the number of feedback bits B. However, because the codebook generation

can be performed offline as long as the correlations are known beforehand, the

complexity of the algorithm is not an issue. We also note that the algorithm

converges to a maximum that is not guaranteed to be global. Nevertheless, it

provides a practical way for codebook design even when the statistics of the

source are not known or difficult to characterize.

2.3.7.2 Multi-antenna receiver, N > 1

If the receiver has multiple antennas (N > 1), the beamforming techniques

discussed for the case of single-antenna receivers could be used, and the re-

ceived signal could be coherently combined across the N antennas. For i.i.d.

Rayleigh channels with M > 1 and N > 1, the Grassmannian solution has

been shown to maximize the beamforming rate [70] [71] [72].

In order to exploit the potential of spatial multiplexing, precoding matrices

with rank J > 1 (and J ≤ min(M,N)) could be used. It is common to

use multidimensional eigenbeamforming, where the columns of the precoding


matrix G ∈ CM×J are orthogonal such that GHG is a diagonal matrix. In

doing so, the J streams are transmitted on mutually orthogonal subspaces, as

is the case when precoding to achieve closed-loop MIMO capacity. We assume

the symbols of the data vector u ∈ CJ are independent and normalized such

that E(uuH

)= IJ . Because the transmit power is trE

(GuuHGH

)= P ,

we have that GHG = diag(P1, . . . , PJ), where Pj (j = 1, ..., J) is the power

allocated to stream j, and∑J

j=1 Pj = P .

Compared to transmitting with equal power on each stream, non-uniform

power allocation requires more feedback and may not result in significant

performance gains, especially if the channel is spatially uncorrelated. As de-

scribed in Chapter 7, spatial multiplexing in 3GPP standards is achieved

using codebook-based precoding with equal power on each stream.

A special case of multidimensional eigenbeamforming is to use antenna

subset selection, where the columns of the precoder G are uniquely drawn

from the columns of the M×M identity matrix and appropriately normalized

[73]. In doing so, J ≤ min(M,N) streams are uniquely associated with a

subset of J transmit antennas. The case of J = M corresponds to the V-

BLAST transmission.

2.4 Practical considerations

2.4.1 CSI estimation

In deriving the MIMO capacity and capacity-achieving techniques, we have

assumed that the CSI is known perfectly at the receiver and, when necessary

for closed-loop MIMO, at the transmitter. In practice, estimates of the CSI

at the receiver can be obtained from training signals (also known as pilot or

reference signals) sent over time or frequency resources that are orthogonal to

the data signals’ resources. ForM transmit antennas, the optimal training set

consists of M mutually orthogonal signals, one assigned for each antenna and

with equal power [74]. The reliability of the CSI estimates and the resulting

rate performance depend on the fraction of resources devoted to the training

signals and the rate of channel variation. As the channel varies more rapidly,

additional training resources are required to achieve the same reliability in the

channel estimates. (Reference signals are described in more detail in Section

4.4.)

2.5 Summary 77

To achieve closed-loop MIMO capacity, the CSI at the transmitter is as-

sumed to be known ideally. If the CSIT is unreliable (see Section 4.4 for

acquiring CSIT), the performance will be degraded. In high SNR channels,

isotropic transmission should be used as an alternative because it requires no

CSIT. In low SNR channels, precoding with limited feedback could be used

to provide performance that is more robust to unreliable CSIT.

2.4.2 Spatial richness

The numerical results in this chapter assume that the MIMO channel coeffi-

cients are i.i.d. Rayleigh. As mentioned in Section 2.1, the spatial correlation

between antennas depends on their spacing relative to the height of the sur-

rounding scatterers. For a base station antenna that is high above the clutter

(for example in rural or suburban deployments), a common rule of thumb is

that spatial decorrelation could be achieved if the separation is at least 10

wavelengths [75]. On the other hand, if the base antenna is surrounded by

scatterers of the same height (for example in rooftop urban deployments),

decorrelation could be achieved with separation of only a few wavelengths.

Decorrelation between pairs of base station antenna elements can also be

achieved using cross-polarized antennas (Section 5.5.1). Mobile terminals are

typically assumed to be surrounded by scatterers, and half-wavelength sepa-

ration is considered sufficient for decorrelation [76].

Full multiplexing gain can be achieved over a SU-MIMO channel if the

antenna array elements at both the transmitter and receiver are uncorrelated.

As the antennas at either the transmitter or receiver become more correlated,

the average capacity of the channel decreases. If the antennas at either end

become fully correlated, then the channel cannot support multiplexing, and

the multiplexing gain is 1. General characterizations of the capacity as a

function of antenna correlation are given in [77].

2.5 Summary

Multiple-antenna techniques can be used to improve the throughput and

reliability of wireless communication. In this chapter, we discussed the single-


user (M,N) MIMO link where the transmitter is equipped with M antennas

and the receiver is equipped with N antennas.

• The open-loop and closed-loop capacity measure the maximum rate of

arbitrarily reliable communication for the case where the channel state

information (CSI) is respectively known and not known at the transmitter.

CSI at the receiver is always assumed.

• The MIMO capacity in a spatially rich channel scales linearly with the

number of antennas. At high SNRs, a multiplexing gain of min(M,N) is

achieved by transmitting multiple streams simultaneously from multiple

antennas. At low SNRs, a power gain of N is achieved through receiver

combining. CSIT for closed-loop MIMO allows more efficient power distri-

bution, resulting in a higher capacity.

• Closed-loop MIMO capacity can be achieved using linear precoding and

linear combining at the transmitter and receiver, respectively, where the

transformations are based on the singular-value decomposition of the chan-

nel matrix H. Waterfilling is used to determine the optimal power alloca-

tion for each of the streams.

• Open-loop MIMO capacity for an (M,N) link (with M ≤ N) can be

achieved using isotropic transmission and an MMSE-SIC receiver. The V-

BLAST transmit architecture achieves capacity by sending independent

streams with appropriate rate assignment on each antenna. The D-BLAST

transmit architecture achieves capacity by cyclically shifting the associa-

tion of streams with transmit antennas.

• Space-time coding provides diversity gain for open-loop MISO channels.

The Alamouti space-time block code achieves the capacity of a (2,1) MISO

channel, but otherwise space-time coding cannot achieve the capacity of

general MIMO antenna configurations. Space-time coding does not provide

multiplexing gain and therefore provides only modest throughput gains as

a result of diversity.

• In FDD systems where CSI at the transmitter is not readily available,

feedback from the receiver can be used to index a fixed set of precoding

matrices, providing suboptimal performance compared to the closed-loop

MIMO capacity.

Chapter 3

Multiuser MIMO

Whereas in the previous chapter we considered a single-user MIMO link

where data streams intended for a single user are multiplexed spatially, in this

chapter we consider multiuser MIMO channels where data streams to/from

multiple users are multiplexed in this way. In particular, we consider two

multiuser Gaussian MIMO channels that are relevant for cellular networks.

The uplink can be modeled by the Gaussian MIMO multiple-access channel

(MAC), a many-to-one network where independent signals from K > 1 users

are received by a single base receiver. The downlink can be modeled by the

Gaussian MIMO broadcast channel (BC), a one-to-many network where a

single base transmitter serves K > 1 users. To avoid needless repetition, we

will henceforth drop the “Gaussian MIMO” qualifier when referring to these

channels.

This chapter focuses on the K-dimensional capacity regions of the MAC

and BC and describes techniques for achieving optimal rate vectors that

maximize a weighted sum-rate metric. As we will see in Chapter 5, this metric

is relevant for scheduling and resource allocation in cellular networks.

3.1 Channel models

The channel models for the MAC and BC were given in Chapter 1, but we

repeat them here for convenience. A ((K,N),M) MAC has K users, each

with N antennas, whose signals are received by a base with M antennas.

The baseband received signal is


793

80 3 Multiuser MIMO

antennas antennas

ll

l

l

l

Multiple-Access Channel(MAC)

User1

UserK

l

l

l

l l

antennas antennas

User1

UserK

Broadcast Channel

(BC)

Fig. 3.1 ((K,N),M) multiple-access channel. K users, each with N antennas, transmit

signals that are demodulated by a receiver with M antennas. (M, (K,N)) broadcast chan-

nel. Transmitter with M antennas serves K users, each with N antennas

.

x =

K∑k=1

Hksk + n, (3.1)

where

• x ∈ CM×1 is the received signal whose mth element (m = 1, . . . ,M) is

associated with antenna m

• Hk ∈ CM×N is the channel matrix for the kth user (k = 1, . . . ,K) whose

(m,n)th entry (m = 1, . . . ,M ;n = 1, . . . , N) gives the complex amplitude

between the nth transmit antenna of user k and the mth receive antenna

3.2 Multiple-access channel (MAC) capacity region 81

• sk ∈ CN×1 is the transmitted signal vector from user k. The covariance is

Qk := E(sks

Hk

), and the signal is subject to the power constraint trQk ≤

Pk with Pk ≥ 0

• n ∈ CM×1 is the additive noise and is a zero-mean spatially white (ZMSW)

additive Gaussian random vector with variance σ2

The components of each transmitted signal vector for the kth user sk (k =

1, . . . ,K) are functions of the information bit stream for the kth user{d(i)k

}.

The receiver demodulates x and provides estimates for these data streams{d(i)k

}.

A (M, (K,N)) BC denotes a base station with M transmit antennas serv-

ing K users, each with N antennas. The base transmits a common signal s

which contains the encoded symbols of the K users’ data streams. This signal

travels over the channel Hk to reach user k (k = 1, . . . ,K). The baseband

received signal by the kth user can be written as:

xk = HHk s+ nk, (3.2)

where

• xk ∈ CN×1 whose nth element (n = 1, . . . , N) is associated with antenna

n of user k

• HHk ∈ C

N×M is the channel matrix for the kth user (k = 1, . . . ,K) whose

(n,m)th entry (n = 1, . . . , N ;m = 1, . . . ,M) gives the complex amplitude

between the nth transmit antenna of user k and the mth receive antenna

• s ∈ CM×1 is the transmitted signal vector which is a function of the data

signals for the K users. The covariance is Q := E(ssH), and the signal is

subject to the power constraint trQ ≤ P , with P ≥ 0

• nk ∈ CN×1 is the additive noise at user k and is a zero-mean spatially

white (ZMSW) additive Gaussian random vector with variance σ2

The components of the transmitted signal s are functions of the K users’

information bit streams{d(i)k

}for k = 1, . . . ,K. The kth user provides esti-

mates for its data stream{d(i)k

}given its received signal xk.

3.2 Multiple-access channel (MAC) capacity region

For a ((K,N),M) MAC, we define a K-dimensional capacity region

82 3 Multiuser MIMO

CMAC

(H1, ...,HK ,

P1

σ2, ...,

PK

σ2

)(3.3)

consisting of achievable rate vectors (R1, . . . , RK), where Rk ≥ 0 for k =

1, . . . ,K. For the point-to-point single-user channel, the capacity C gives the

fundamental performance limit, in that reliable communication is possible

for any rate R < C and impossible for any rate R > C. For the MAC

capacity region, any vector that belongs to the region CMAC is a K-tuple

of rates that can be simultaneously achieved by the users. An important

scalar performance metric derived from the capacity region is the sum-rate

capacity (or sum-capacity), which is the maximum total throughput that can

be achieved:

maxR∈CMAC

K∑k=1

Rk. (3.4)

In the remainder of this section, we characterize the capacity region for

the MIMO MAC channel (3.1), starting with the single-antenna MAC. We

describe how to achieve rate vectors lying on the boundary of the capacity

region. A formal derivation of the MAC capacity region is beyond the scope

of this book, but those interested can refer to [37].

3.2.1 Single-antenna transmitters (N = 1)

We consider the special case of (3.1) for the ((2, 1), 1) MAC channel where

signals fromK = 2 mobiles, each with a single antenna, are received by a base

with a single antenna (M = 1). We let hk (k = 1, 2) be the time-invariant

SISO channel for each user, and we assume that these values are known at the

base. The capacity region, shown in Figure 3.2 for fixed channel realizations

h1 and h2, is the set of non-negative rate pairs (R1, R2) that satisfies the

following three constraints:

R1 ≤ log2

(1 +

P1

σ2|h1|2

)

R2 ≤ log2

(1 +

P2

σ2|h2|2

)

R1 +R2 ≤ log2

(1 +

P1

σ2|h1|2 + P2

σ2|h2|2

). (3.5)


Figure 3.2 illustrates this capacity region. Point A on the capacity region

boundary can be achieved by having user 1 transmit with a Gaussian code-

word with power P1 and having user 2 be silent. Similarly, point B is achieved

with user 2 on and user 1 off. Each rate pair on the line segment joining A

and B can be achieved by time-multiplexing the transmissions between the

two users with an appropriate fraction of time for each user. The triangular

region bounded by this segment is called the time-division multiple access

(TDMA) rate region.

While time-multiplexing of users for the MAC seems like a reasonable

strategy for achieving high throughput, it turns out we can do better by hav-

ing the users transmit simultaneously. For example, point C on the boundary

of the MAC capacity region can be achieved by having both users transmit

with full power. The signal s2 for user 2 is decoded in the presence of inter-

ference from user 1’s signal and thermal noise. The rate achieved by user 2

is

R2 = log2

(1 +

P2|h2|2σ2 + P1|h1|2

). (3.6)

Since h2 and s2 are known at the receiver, the received data signal for user

2 can then be reconstructed and subtracted from the received signal. The

signal for user 1 can then be decoded in the presence of only thermal noise,

achieving rate

R1 = log2

(1 +

P1

σ2|h1|2

). (3.7)

Adding the rates from (3.6) and (3.7), the sum rate is:

R1 +R2 = log2

(1 +

P1

σ2|h1|2 + P2

σ2|h2|2

),

thereby achieving the sum-rate capacity (3.5). By switching the decoding

order, one can achieve the same sum rate but at point D. To summarize,

the sum-rate capacity is achieved by having both users transmit with full

power and using interference cancelation at the receiver. The order of the

cancelation does not affect the sum rate, and any point between the vertices

C and D can be achieved with time-sharing between the two decoding orders.

Generalizing to the K-user case, the K-dimensional capacity region for the

((K, 1), 1) MAC is bounded by 2K − 1 hyperplanes, each corresponding to

a nonempty subset of users. The capacity region is a generalized pentagonal

region known as a polymatroid [78]. The sum-rate capacity is achieved by

having all users transmit with full power and using successive interference

84 3 Multiuser MIMO

cancelation, where users are successively detected and canceled one by one

in the presence of noise and interference from uncanceled users. As in the 2-

user case, the sum-rate capacity does not depend on the cancelation order. In

general, the sum-rate capacity region hasK! corner points, and these are each

achieved using a different cancelation order. Generalizing further to multiple

receive antennas (M > 1), the ((K, 1),M) MAC capacity region is

CMAC(h1, . . . ,hK , P1

σ2 , . . . ,PK

σ2 ) ={(R1, . . . , RK) :

∑k∈S

Rk ≤ log2 det

(IM +

∑k∈S

Pk

σ2 hkhHk

), ∀S ⊆ {1, ...,K}

},

(3.8)

where there is one rate constraint for each nonempty subset S of users. Note

that if the transmit powers of the users are all equal (P1 = P2 = . . . = PK =

P/K), then the sum-capacity of the ((K, 1),M) MAC is equivalent to the

(K,M) SU-MIMO open-loop capacity with power constraint P where the

columns of the M×K channel are the K user’s channels (H = [h1, . . . ,hK ]).

Therefore, since the MMSE-SIC receiver achieves the SU-MIMO open-loop

capacity (Section 2.3.3), it can be used to achieve the sum-capacity of the

((K, 1),M) MAC where the users transmit with equal power. More generally,

the MMSE-SIC receiver can achieve the MAC sum-capacity for any set of

powers P1, . . . , PK .

(bps/Hz)

(bps

/Hz)

DB

C

A

Fig. 3.2 ((2,1),1) MAC capacity region: K = 2 users, each with N = 1 antenna, and

a receiver with M = 1 antenna. The TDMA rate region is bounded by the line segment

joining points A and B.


3.2.2 Multiple-antenna transmitters (N > 1)

If the users have multiple antennas (N > 1), the transmitted signal from user

k is an N -dimensional vector sk ∈ CN×1 with covariance Qk. Generalizing

the capacity region of a ((K, 1),M) MAC (3.8), the capacity region of a

((K,N),M) MAC with fixed covariances Q1, . . . ,QK is

CMAC({Hk} ,{Qk/σ

2}) =⎧⎪⎨

⎪⎩(R1, . . . , RK) :∑k∈S

Rk ≤ log2 det

(IM +

1

σ2

∑k∈S

HkQkHHk

), ∀S ⊆ {1, ...,K}

⎫⎪⎬⎪⎭ .

(3.9)

The shaded pentagonal region of Figure 3.3 shows an example of the ca-

pacity region for a ((2, N),M) MAC with fixed covariances Q1 and Q2, each

with rank r = 2. The transmitted signal for user k (k = 1, 2) can be written

as sk = Gkuk, where Gk ∈ CN×r is the precoding matrix, and uk ∈ C

r×1

is the data vector. Point A on the capacity region boundary can be achieved

using an MMSE-SIC as depicted in Figure 3.4 where the streams of user

1 are detected first. By treating the transmitted signal of user k as a lin-

ear combination of r = 2 independent data streams, the ((2, N),M) MAC

is equivalent to a MAC with four virtual users, each transmitting a single

stream. The rates of the two users at point A are respectively

R1 = log2det(IM + 1

σ2H1Q1HH1 + 1

σ2H2Q2HH2

)det(IM + 1

σ2H2Q2HH2

)and

R2 = log2 det

(IM +

1

σ2H2Q2H

H2

).

For a different set of covariances Q1 and Q2 (subject to the power constraints

trQ1 = P1 and trQ2 = P2), the capacity region will correspond to a differ-

ent pentagonal region as indicated by the dashed boundaries in Figure 3.3.

Taking the union of all such capacity regions yields the ((2, N),M) MAC ca-

pacity region CMAC(H1,H2, P1/σ2, P2/σ

2) whose boundary is indicated by

the heavy solid line in Figure 3.3.

In general, the ((K,N),M) MAC capacity region can be written as the

union of capacity regions (3.9) with different covariances:

86 3 Multiuser MIMO

CMAC({Hk} ,{Pk/σ

2}) =

⋃Qk≥0

trQk≤Pk

⎧⎪⎨⎪⎩

(R1, . . . , RK) :∑k∈S

Rk ≤ log2 det

(IM +

1

σ2

∑k∈S

HkQkHHk

), ∀S ⊆ {1, ...,K}

⎫⎪⎬⎪⎭ .

(3.10)

A capacity-achieving architecture for the MAC is shown in Figure 3.5.

User k demultiplexes its information bits{d(i)k

}into rk ≥ 1 parallel streams,

and then encodes and modulates each of these streams. The symbol vector

uk ∈ Crk is weighted by a precoding matrix Gk ∈ C

M×rk such that the trans-

mitted signal is sk = Gkuk. How are the transmit covariances Q1, . . . ,QK

determined to achieve a particular rate vector on the capacity region bound-

ary? Recall that for the single-user MIMO channel, the capacity-achieving

covariance Q is based on the SVD of the MIMO channel H. For the MAC,

if all the users’ MIMO channels are mutually orthogonal (i.e., HHi Hj = 0M ,

for all i �= j), then the capacity-achieving covariances for each user individ-

ually also achieve the MAC capacity region boundary. In general, however,

because of the interaction of the users’ MIMO channels, the MAC capacity

region is not achieved using the optimal single-user covariances. Determining

the capacity-achieving covariances for the MAC is not easy because of the

channel interactions. While there is no known general closed-form expres-

sion for computing the optimal covariances for achieving the capacity region

boundary, one can compute them numerically using iterative techniques de-

scribed in Section 3.5.1.

For a given set of covariances Q1, . . . ,QK and a given decoding order for

the users, the MMSE-SIC receiver successively decodes and cancels users so

that each user’s SINR is maximized in the presence of noise and interfer-

ence from all other users not yet decoded. Without loss of generality, we can

assume that users are decoded in order from user 1 up to user K. If the can-

cellation is ideal, then in decoding the data stream for user k (k = 1, . . . ,K),

the received signal x (3.1) has been cleansed of the interference from users

1, . . . ,K − 1. Its virtual received signal is

x−k−1∑j=1

Hjsj =K∑

j=k

Hjsj + n, (3.11)

so sk is received in the presence of interference from users k + 1, . . . ,K.

Proceeding as in Section 2.3.2, we can see that user k achieves rate

3.3 Broadcast channel (BC) capacity region 87

Rk = log2

det(IM + 1

σ2

∑Kj=k HjQjH

Hj

)det(IM + 1

σ2

∑Kj=k+1 HjQjHH

j

) (3.12)

= log2 det

⎡⎢⎣IM +

⎛⎝σ2IM +

K∑j=k+1

HjQjHHj

⎞⎠−1

HkQkHHk

⎤⎥⎦ (3.13)

by whitening its virtual received signal (3.11) with respect to the noise and in-

terference covariance and then performing MMSE-SIC for its desired symbol

vector uk.

A

A

(bps/Hz)

(bps

/Hz)

Fig. 3.3 The shaded pentagonal rate region is for a K = 2 MAC with fixed covariances

Q1 and Q2, with trQk ≤ Pk. The general MAC capacity region is the union of all such

pentagonal rate regions for different covariances that satisfy the power constraints.

3.3 Broadcast channel (BC) capacity region

For a (M, (K,N)) BC, we define the K-dimensional capacity region

CBC(HH

1 , ...,HHK , P/σ2

)(3.14)

as the set of rate vectors (R1, . . . , RK) that can be simultaneously achieved

by the users. Whereas in the MAC the power of each user is individually

constrained, in the BC, the transmitter is able to partition power among the

88 3 Multiuser MIMO

MMSE receiver 1a (w.r.t. interf. 1b, 2a, 2b) Decode 1a

Cancel 1a MMSE receiver 1b (w.r.t. interf. 2a, 2b)

Decode 1b

Cancel 1b

Cancel 2a

Decode 2a

Decode 2b

MMSE receiver 2a (w.r.t. interf. 2b)

MF receiver 2b

(received signal)Stream 1a

Stream 1b

Stream 2a

Stream 2b

Fig. 3.4 An illustration of the MMSE with successive interference cancellation for achiev-

ing point A in the 2-user rate region of Figure 3.3

.

M

MMux,

coding,modulation

Mux, coding,

modulation

�

Pre-coding

Δ

lM MMSE-SIC

Pre-coding

Δ

Fig. 3.5 A capacity-achieving strategy for the MIMO MAC. User k uses matrix Gk to

precode its data vector uk. A MMSE-SIC is used at the receiver to jointly detect and

demodulate the user data streams.

multiple users so the total power allocated to all users is constrained by P .

In this section, we characterize the BC capacity region and discuss capacity-

achieving techniques. As we will see, rate vectors on the BC capacity region

boundary can be achieved using a transmitter encoding technique that is

analogous to the capacity-achieving successive interference cancellation used

for the MAC.


3.3.1 Single-antenna transmitters (M = 1)

We first consider the case of a (1, (2, N)) broadcast channel with a single-

antenna transmitter serving two users each with N ≥ 1 antennas. Suppose

the M × 1 SIMO channels hH1 and hH

2 are fixed such that ||h1|| < ||h2||. ForM = 1, the broadcast channel is degraded, meaning that the users’ channels

can be ordered so that the received signal at a user can be regarded as a more

noisy version of the signal received by the previous user in the ordering. In

this case, we can imagine that user 1 is farther from the transmitter than user

2 and that it receives a further degraded version of the signal transmitted

with power P . The capacity region of the 2-user degraded broadcast channel

is given by the rate pairs (R1, R2) such that

R1 ≤ log2

(1 +

P1||h1||2σ2 + P2||h1||2

)(3.15)

R2 ≤ log2

(1 +

P2

σ2||h2||2

), (3.16)

where P1 and P2 are the non-zero powers allocated to the two users that

satisfy the power constraint P1 + P2 ≤ P .

The rate vector corresponding to a given power allocation P1 and P2 can be

achieved by encoding the data signals u1 and u2 for the users with respective

powers P1 = E(|u1|2

)and P2 = E

(|u1|2). The transmitted signal is the

superposition or sum of the codewords, s = u1 + u2. User 1 detects its signal

in the presence of noise and interference u2 to achieve rate R1 given by (3.15).

Because user 2 has a higher channel gain than user 1, it could detect u1, cancel

it, and detect u2 in the presence of only AWGN.

Whereas superposition coding as above relies on interference cancelation at

the receiver, an alternative technique for achieving the capacity region places

the burden of complexity on the transmitter. This technique, known as dirty

paper coding (DPC) [79], is an encoding process that occurs jointly and in an

ordered fashion among the users so that a given user experiences interference

only from users encoded after it. In other words, a given user receives zero

power from the signals of users that are encoded earlier in the sequence,

as if the signals of these users were “pre-subtracted” at the transmitter. In

our example, user 1 would be encoded in a conventional (capacity-achieving)

manner, then user 2 would be encoded using DPC with non-causal knowledge

of user 1’s data signal.

90 3 Multiuser MIMO

The left subfigure of Figure 3.6 shows the 2-user capacity region for ||h1|| <||h2||, obtained from (3.15) and (3.16) by varying the power partition between

the users. We note that by switching the DPC encoding order so that user

2 is encoded first, user 1 is decoded without interference, and the resulting

rate region is concave.

For the general K-user degraded broadcast channel (still with a single-

antenna transmitter), we assume without loss of generality that the channels

are ordered ||h1|| ≤ ||h2|| ≤ . . . ≤ ||hK ||. Under superposition coding, the kth

user detects and cancels the signals from users 1, 2, . . . , k− 1 whose channels

are weaker. Under DPC, the users are encoded in order from user 1 to 2 up

to K. The rate region is given by the rate vectors R1, . . . , RK satisfying

CBC(h1, . . . ,hK , P ) ={(R1, . . . , RK) : Rk ≤ log2

(1 +

Pk||hk||2σ2 +

∑Kj=k+1 Pj ||hk||2

),

K∑k=1

Pk ≤ P

}.

(3.17)

Encode user 2 then user 1


(bps/Hz)

(bps

/Hz)

Encode user 1then user 2

(bps/Hz)

(bps

/Hz)

A

B


Convex hull

Fig. 3.6 The left subfigure shows the 2-user capacity region for a degraded BC with

M = 1. The right subfigure shows the 2-user capacity region for a non-degraded BC with

M > 1. This region is the convex hull of the union of rate regions for the two encoding

orders.


3.3.2 Multiple-antenna transmitters (M > 1)

In the case of multiple-antenna transmitters, M > 1, the users’ channels can

no longer be absolutely ranked in terms of their channel strengths because the

additional degrees of freedom allow the transmitter to spatially steer power

between users. For example, consider the case of two single-antenna users with

1×M MISO channels hH1 and hH

2 such that ||h1|| ≤ ||h2||. We could weight

the codewords u1 and u2 for users with respective M -dimensional unit-power

beamforming vectors g1 and g2 so that |hH1 g1| > |hH

2 g2|, which is opposite to

the MISO channel ranking. Since an absolute ranking of the user channels no

longer exists, the broadcast channel for M > 1 is said to be non-degraded. For

such channels, superposition coding is no longer capacity-achieving; however,

DPC achieves the capacity region of the MIMO BC [80].

Let us consider a (4,(2,1)) broadcast channel with a M = 4-antenna trans-

mitter serving two single-antenna users. We assume that the channels h1

and h2, and beamforming vectors g1 and g2, are fixed. If user 1 is encoded

first, the transmitter first picks a codeword u1. The transmit covariance is

Q1 := g1gH1 E(||u1||2

). With knowledge of u1, Q1, and h2, the transmitter

picks codeword u2 using DPC. The transmit covariance for user 2 is Q2, and

the total transmit power is constrained so that tr(Q1) + tr(Q2) ≤ P . As a

result of DPC, the rate pairs (R1, R2) satisfy

R1 ≤ log2

(1 + 1

σ2hH1 (Q1 +Q2)h1

1 + 1σ2hH

1 Q2h1

)(3.18)

R2 ≤ log2

(1 +

1

σ2hH2 Q2h2

). (3.19)

Switching the encoding order, the rate pairs satisfy

R1 ≤ log2

(1 +

1

σ2hH1 Q1h1

)(3.20)

R2 ≤ log2

(1 + 1

σ2hH2 (Q1 +Q2)h2

1 + 1σ2hH

2 Q1h2

). (3.21)

The right subfigure of Figure 3.6 shows the rate regions achieved with the

two encoding orders, where different vectors on the boundary are achieved by

varying the power split between the two users. Unlike the degraded BC case,

the regions are neither convex nor concave, and the union of these regions

is not convex. The broadcast channel capacity region is the convex hull (in

other words, the minimal convex set) of the union of these two rate regions.

92 3 Multiuser MIMO

Rate vectors in the convex hull that lie outside these two rate regions (for

example, those vectors on the segment between points A and B in the right

subfigure of Figure 3.6) can be achieved using time-sharing.

In general, for multiple transmit antennas M > 1 and multiple receive

antennas N > 1, the broadcast channel capacity region is the convex hull of

the union of rate regions corresponding to all possible encoding permutations:

CBC({

HHk

}, P/σ2

)= Co

(⋃πRπ

), (3.22)

where Co denotes the convex hull operation and Rπ denotes the rate region

for a given encoding permutation π. This rate region is given as

Rπ =

{(Rπ(1), . . . , Rπ(K)) : Rπ(k) =

log2

det(IN + 1

σ2HHπ(k)

∑Kj=k Qπ(j)Hπ(k)

)det(IN + 1

σ2HHπ(k)

∑Kj=k+1 Qπ(j)Hπ(k)

) , K∑j=1

tr (Qj) ≤ P

},(3.23)

where π(k) (k = 1, . . . ,K) denotes the index of the kth encoded user.

For a given permutation π and a given set of transmit covariances, the

received signal by user π(k) experiences no interference from those users

encoded before it (π(1), . . . , π(k − 1)) as a result of DPC. Therefore from

(3.2), its virtual received signal is

xπ(k)= HH

π(k)s+

K∑j=k+1

HHπ(j)

s+ nπ(k), (3.24)

where the interference is caused by signals intended for users π(k+1), . . . , π(K).

From (3.23), user π(k) can achieve a rate of

Rπ(k) = log2

det(IN + 1

σ2HHπ(k)

∑Kj=k Qπ(j)Hπ(k)

)det(IN + 1

σ2HHπ(k)

∑Kj=k+1 Qπ(j)Hπ(k)

)

= log2 det

⎡⎢⎣IN+

⎛⎝σ2IN +

K∑j=k+1

HHπ(k)Qπ(j)Hπ(k)

⎞⎠−1

HHπ(k)Qπ(k)Hπ(k)

⎤⎥⎦

by whitening its virtual received signal (3.24) with respect to its noise and

interference covariance and then performing MMSE-SIC with respect to its

desired symbol vector uπ(k). Figure 3.7 shows a block diagram for achieving

3.4 MAC-BC Duality 93

the BC capacity region (3.22) using DPC and precoding at the transmitter

and MMSE-SIC at each receiver.

Dirty paper coding

Pre-coding

Δ

Pre-coding

Δ

�

�

�

�

MMSE-SIC

Σ

MMSE-SIC

�

Fig. 3.7 A capacity-achieving strategy for the MIMO BC. The transmitter uses dirty

paper encoding to jointly encode the users’ data streams. The symbol vector uk for user

k is precoded using the matrix Gk. At the receiver for user k, the signal is whitened with

respect to interference from signals not eliminated from DPC, and MMSE-SIC is performed

with respect to the desired symbol vector uk.

3.4 MAC-BC Duality

In this section, we describe a useful duality [81] between the MIMO MAC

capacity region (3.10) and the MIMO BC capacity region (3.22) that provides

valuable insights and a tool for evaluating the performance of a MIMO BC

channel (described in Section 3.5). For a given (M, (K,N)) MIMO BC where

the kth user’s channel is an N×M matrix HHk , we define a dual ((K,N),M)

MIMO MAC where the kth user’s channel is the M × N matrix Hk. The

following MAC-BC duality characterizes the MIMO BC capacity region in

terms of the dual MIMO MAC capacity region:

CBC({

HHk

}, P/σ2

)=

⋃Pk≥0,

∑Kk=1 Pk≤P

CMAC({Hk} ,

{Pk/σ

2})

. (3.25)

It states that any rate vector in the (M, (K,N)) BC capacity region with

power constraint P can be obtained on the dual ((K,N),M) MAC if the

mobiles are allowed to distribute power P among themselves. As an example,

Figure 3.8 shows the BC and dual MAC capacity regions for K = 2, M =

94 3 Multiuser MIMO

N = 1. The BC rate region with P = 2 is the union of the dual MAC capacity

regions where the power is split between the users.

(bps/Hz)

(bps

/Hz)

A

:

:

Fig. 3.8 K = 2-user BC capacity region, shown as the union of the dual MAC capacity

regions with different power splits between the users.

The proof of the duality relies on a transformation between the covari-

ances for the MAC and the BC, where, for any set of MAC covariances

QMAC1 , ...,QMAC

K of size N × N and any decoding order for the users, there

exists a set of BC covariances QBC1 , ...,QBC

K of size M×M with the same sum

power such that the rates achieved by each of the K users are the same. The

converse also holds, so that for any set of BC covariances and any encoding

order, there exists a set of MAC covariances with the same sum power such

that the user rates are the same.

In particular, the transformation relies on a reversal of the decoding and

encoding orders. For example, if users on the MAC are decoded in order start-

ing with user 1 and ending with user K (so that user K sees no interference),

then the users on the BC are encoded in reverse order from user K to user 1

(so that user 1 sees no interference). With this decoding order, the rate for

user k on the MAC is

3.5 Scalar performance metrics 95

RMACk = log2

det(IM + 1

σ2

∑Kj=k HjQ

MACj HH

j

)det(IM + 1

σ2

∑Kj=k+1 HjQMAC

j HHj

) . (3.26)

And the rate for user k on the BC is

RBCk = log2

det(IN + 1

σ2HHk

∑kj=1 Q

BCj Hk

)det(IN + 1

σ2HHk

∑k−1j=1 Q

BCj Hk

) . (3.27)

Using the duality transformation, the sum powers are the same:

K∑k=1

trQBCk =

K∑k=1

trQMACk , (3.28)

and user rates are the same: RMACk = RBC

k , k = 1, 2, ...,K.

3.5 Scalar performance metrics

Given a K-dimensional capacity region CMAC or CBC, it would be useful to

define scalar performance metrics that map the multidimensional region to a

single quantity in order to make performance comparisons between different

strategies. The sum-rate capacity is one such metric which we defined for the

MAC (3.4), and we could define it similarly for the BC capacity region:

maxR∈CBC

K∑k=1

Rk. (3.29)

More generally, we could define a weighted sum-rate metric

maxR∈C

K∑k=1

qkRk, (3.30)

where C denotes either the MAC or BC capacity region and where the weights

are non-negative constants qk ≥ 0, k = 1, . . . ,K. We will see in Section 5.4.2

how these weights can be set to achieve quality-of-service requirements in the

context of scheduling and resource allocation. For this reason, we refer to the

weights q1, . . . , qK as quality-of-service (QoS) weights. A user with a higher

QoS weight has higher relative importance in the sense that it contributes

more to the weighted sum-rate metric. In practice, we are interested not in

96 3 Multiuser MIMO

the weighted sum rate but in the rate vector which maximizes the weighted

sum rate:

ROpt = argmaxR∈C

K∑k=1

qkRk. (3.31)

Because the capacity regions are convex and the weights are nonnegative, the

desired rate vector ROpt lies on the boundary of the rate region.

Note that as a consequence of the MAC-BC duality, given P and a set

of QoS weights, the BC weighted sum rate is an upper bound on the MAC

weighted sum rate for any power partition P1, . . . , PK such that∑K

k=1 Pk =

P :

maxR∈CBC({HH

k },P/σ2)

K∑k=1

qkRk ≥ maxR∈CMAC({Hk},{Pk/σ2})

K∑k=1

qkRk. (3.32)

In the context of a single base station serving multiple users, if the base

transmit power is equal to the sum of the user transmit powers, the downlink

performance will be at least as good as the uplink performance using the

reciprocal channels.

3.5.1 Maximizing the MAC weighted sum rate

Achieving the optimal rate vector that maximizes the weighted sum rate

(3.31) is straightforward for the MAC with fixed covariances because, as a

result of the polymatroid shape of the rate region, the optimal rate vector

is one of the vertices of the sum-rate capacity hyperplane. To lie on this

hyperplane, all users transmit with a fixed covariance at full power, and the

desired vertex is given by the cancelation order related to the ranking of the

QoS weights [78]. Without loss of generality, if q1 ≤ q2 ≤ . . . ≤ qK , then

the optimal rate vector ROpt which maximizes (3.31) is given by the vertex

where the users are decoded in order from 1 to K so that user 1 experiences

interference from all other users, and user K experiences no interference. The

rate Rk for user k (k = 1, . . . ,K) is given by (3.12).

While a network operator may be interested in determining the rate vector

R that maximizes the weighted sum rate, the actual measured performance

in terms of throughput is given by the sum rate∑K

k=1 Rk. In summing these

rates given by (3.12), the denominator for Rk (k = 1, . . . ,K−1) cancels with

the numerator for Rk+1. The sum rate with fixed covariances is therefore


given just by the numerator of R1:

maxR∈CMAC({Hk},{Qk/σ2})

K∑k=1

Rk = log2 det

⎛⎝IM +

1

σ2

K∑j=1

HjQjHHj

⎞⎠ . (3.33)

Note that while the individual user rates in (3.12) were determined by the

decoding order, the sum rate (3.33) is independent of the ordering because

any vertex maximizing the weighted sum rate lies on the sum-rate capacity

hyperplane.

We can visualize this concept for the K = 2-user MAC rate region shown

in Figure 3.9. For fixed q1 and q2, contour lines for a given weighted sum rate

have slope −q1/q2. Therefore if q2 > q1, the optimal rate vector is achieved by

decoding user 1 first. On the other hand, if q1 > q2, the optimal rate vector

is achieved by decoding user 2 first. In general, the user with the largest QoS

weight is decoded last so that it experiences no interference.

Contour lines for

(bps/Hz)

(bps

/Hz)

Contour lines for

Fig. 3.9 MAC capacity region and contour lines for weighted sum rates given q1 and q2.

The rate vectors that maximize the weighted sum rate are given by ROpt.

To maximize the weighted sum rate when the transmit covariances are

not specified, the optimal decoding order is still given by the ranking of the

QoS weights because this ordering is the best once the covariances are fixed.

If q1 ≤ q2 ≤ . . . ≤ qK , the maximum weighted sum rate is obtained by

optimizing over the transmit covariances, subject to the power constraint for

each user:

98 3 Multiuser MIMO

maxR∈CMAC({Hk},{Pk/σ2})

K∑k=1

qkRk =

maxtrQk≤Pk

K∑k=1

qk log2

det(IM + 1

σ2

∑Kj=k HjQjH

Hj

)det(IM + 1

σ2

∑Kj=k+1 HjQjHH

j

) .(3.34)Iterative algorithms to determine the optimal covariances can be found in [82]

for sum rate maximization, and in [83] for weighted sum rate maximization.

Because the weighted sum rate maximization problem (3.34) is a con-

vex optimization, conventional numerical optimization techniques can also

be used [84]. For example, the following gradient-based optimization simpli-

fies the problem by reducing the multidimensional optimization in each step

to a single-dimensional optimization [85]. We assume without loss of gener-

ality that the QoS weights are ordered: q1 ≤ q2 ≤ . . . ≤ qK . We define the

weighted sum rate objective function as a function of the covariances Q(n)k ,

k = 1, . . . ,K, during iteration n:

f(Q(n)1 , . . . ,Q

(n)K ) :=

K∑k=1

qk log2

det(IM + 1

σ2

∑Kj=k H

Hj Q

(n)j Hj

)det(IM + 1

σ2

∑Kj=k+1 H

Hj Q

(n)j Hj

) (3.35)

= q1 log2 det

⎛⎝IM +

1

σ2

K∑j=1

HHj Q

(n)j Hj

⎞⎠+

K∑k=2

(qk − qk−1) log2 det

⎛⎝IM +

1

σ2

K∑j=k

HHj Q

(n)j Hj

⎞⎠ .

We also define the gradient of the objective function with respect to the ith

covariance Qi, i = 1, . . . ,K:

∇fi(Q(n)1 , ...,Q

(n)K ) := q1

[HH

i

(IM +

1

σ2

∑K

j=1HH

j Q(n)j Hj

)−1

Hi

]+

i∑k=2

(qk − qk−1)

⎡⎢⎣Hi

⎛⎝IM +

1

σ2

K∑j=k

HHj Q

(n)j Hj

⎞⎠−1

HHi

⎤⎥⎦. (3.36)

The following algorithm proceeds iteratively until the covariance matrices

converge to the optimal covariance matrices.


Algorithm for computing optimal MAC covariances

Let n := 1

Initialize Q(n)k := 0, k = 1, . . . ,K

while not converged

for k = 1 to K

a. Let λ2k be the dominant eigenvalue of

∇fk

(Q

(n+1)1 , . . . ,Q

(n+1)k−1 ,Q

(n)k ,Q

(n)k+1, . . . ,Q

(n)K

)and let vk be the corresponding eigenvector

b. Find t∗ which is the solution to the one-dimensional

optimization:

t∗ = argmaxt∈[0,1] f(Q

(n+1)1 , . . . ,Q

(n+1)k−1 ,

tQ(n)k + (1− t)Pkvkv

Hk ,Q

(n)k+1, . . . ,Q

(n)K

)c. Update the kth user’s covariance:

Q(n+1)k := t∗Q(n)

k + (1− t∗)PkvkvHk

end

n := n+ 1

end

3.5.2 Maximizing the BC weighted sum rate

For the MIMO MAC, the weighted sum rate objective is convex with respect

to the user covariances. In contrast, for the MIMO BC, this metric is in

general not convex with respect to the transmit covariances, making the

direct computation of the maximum weighted sum rate difficult. However, as

a result of the MAC-BC duality in Section 3.4, the MIMO BC weighted sum

rate can be characterized as a convex function in terms of the dual MAC

transmit covariances. Using the duality, the maximum weighted sum rate for

the MIMO BC can be written as

maxR∈CBC({Hk},P/σ2)

K∑k=1

qkRk = maxR∈⋃

Pk,∑

Pk≤P CMAC({Hk},{Pk/σ2})

K∑k=1

qkRk

100 3 Multiuser MIMO

= maxPk,

∑Pk≤P

maxtrQk≤Pk

K∑k=1

qk log2

det(IM + 1

σ2

∑Kj=k HjQjH

Hj

)det(IM + 1

σ2

∑Kj=k+1 HjQjHH

j

) . (3.37)A variation of the technique for maximizing the MIMOMAC weighted sum

rate (Section 3.5.1) can be used for the MIMO BC [85]. We use the same ob-

jective function (3.35) and gradient (3.36), where HHk is the M × N dual

MAC channel for the kth user. The covariances Q(n)k are again size N ×N .

Compared to the algorithm for the MAC in Section 3.5.1, the following it-

erative algorithm for the dual MAC allows for power sharing among the users.

Algorithm for computing optimal dual MAC covariances

Let n := 1

Initialize Q(n)k := 0, k = 1, . . . ,K

while not converged

a. For each user k = 1, . . . ,K, let λ2k be the dominant eigenvalue of

∇fk

(Q

(n)1 , . . . ,Q

(n)k−1,Q

(n)k ,Q

(n)k+1, . . . ,Q

(n)K

)and let vk be the corresponding eigenvector

b. Let i∗ = argmax(λ21, . . . , λ

2K)

c. Find t∗ which is the solution to the one-dimensional optimization:

t∗ = argmaxt∈[0,1] f(tQ

(n)1 , . . . , tQ

(n)i∗−1,

tQ(n)i∗ + (1− t)Pkvi∗v

Hi∗ , tQ

(n)i∗+1, . . . , tQ

(n)K

)d. Update the covariances:

Q(n+1)i∗ := t∗Q(n)

i∗ + (1− t∗)Pvi∗vHi∗

Q(n+1)j := t∗Q(n)

j , j �= i∗

e. n := n+ 1

end

After the algorithm converges, the BC covariances (of size M × M) are

computed from the dual MAC covariances (of size N × N) via the duality

transformation [81]. Recall that for a given decoding order of the dual MAC

users, the corresponding BC user rates are achieved by reversing the order

for the downlink encoding. Therefore, given the ordered QoS weights q1 ≤

3.6 Sum-rate performance 101

q2 ≤ . . . ≤ qK , the MAC weighted sum rate is maximized by decoding the

users in order starting with user 1 and ending with user K, and the BC

weighted sum rate is maximized by encoding the users in order starting with

user K and ending with user 1. As highlighted in Figure 3.10, the user with

the highest QoS weight (user K) experiences no interference on the uplink

but experiences interference from all other users on the downlink.

MAC (SIC) decoding order:

BC (DPC) encoding order:

QoS weights:

1 2 K-1 K

K K-1 2 1…

…

Experiences no interference

Experiencesinterferencefrom all other users

Fig. 3.10 To maximize the weighted sum rate for the MAC, users are decoded with

successive interference cancelation (SIC) in the order of ascending QoS weights. For the

BC, users are encoded with dirty paper coding (DPC) in the order of descending QoS

weights.

3.6 Sum-rate performance

In this section, we study the sum-rate capacity performance of the MIMO

MAC and BC. Our observations will provide motivation for designing subop-

timal transceiver techniques (Chapter 5) and provide some insights into the

cellular system performance when users have different geometries (Chapter

6). We use the ((K,N),M) MAC signal model (3.1) and the (M, (K,N)) BC

signal model (3.2). The BC has transmit power P , and each user on the MAC

has transmit power P/K, so that the total power is the same for the two sys-

tems. We use the following notation to denote the sum-rate capacity of the

MAC for channels H1, . . . ,HK and of the BC for channels HH1 , . . . ,HH

K :


CMAC({Hk} , P/σ2

):= max

R∈CMAC({Hk},{P/(Kσ2)})

K∑k=1

Rk (3.38)

CBC({

HHk

}, P/σ2

):= max

R∈CBC({HHk },P/σ2)

K∑k=1

Rk. (3.39)

As an alternative way to communicate over the MIMO BC, we consider time-

division multiple access (TDMA) transmission where users are served one at

a time. With knowledge of CSI at the transmitter, the maximum sum rate for

TDMA is achieved by serving the user with the highest CL-MIMO capacity:

CTDMA({

HHk

}, P/σ2

):= max

kmax

trQ=Plog det

(I+

1

σ2HkQHH

k

). (3.40)

This strategy is also referred to as time-sharing [86].

As another performance benchmark, we consider the open-loop and closed-

loop capacity of a single-user channel H ∈ CM×KN , which is the concatena-

tion of the multiuser channels:

H := [H1, . . . ,HK ] . (3.41)

With power constraint P , we define the capacities of the corresponding single-

user channels as:

COL(H, P/σ2) := log2 det

(IM +

P

σ2KNHHH

)(3.42)

CCL(H, P/σ2) := maxQ>0,trQ=P

log2 det

(IM +

1

σ2HQHH

). (3.43)

Given the sum-rate metrics defined for a given channel realization, we can

define the respective average sum rates where the expectation is taken over

random channel realizations:

CMAC((K,N),M, P/σ2) := E{Hk}[CMAC

({Hk} , P/σ2)]

(3.44)

CBC(M, (K,N), P/σ2) := E{HHk }[CBC

({HH

k

}, P/σ2

)](3.45)

CTDMA(M, (K,N), P/σ2) := E{HHk }[CTDMA

({HH

k

}, P/σ2

)]. (3.46)

In the numerical results that follow, we assume the channels are i.i.d.

Rayleigh.


3.6.1 MU-MIMO sum-rate capacity and SU-MIMO

capacity

For a given set of channels H1, . . . ,HK , we can establish a useful relationship

among the MIMOMAC sum-rate capacity (3.38), MIMO BC sum-rate capac-

ity (3.39), and the capacity of the corresponding SU-MIMO channels (3.43)

and (3.42). Let us first define a ((KN, 1),M) MAC for KN single-antenna

users where the SIMO channels of each user are given by the columns of

[H1, . . . ,HK ] and where the transmit power of each user is P/(KN). We

let CMAC(h11, . . . ,h1N , . . . ,hK1, . . . ,hKN , P/σ2) be the sum-rate capacity

of this MAC. Due to the equivalence between the open-loop MIMO capac-

ity and the MAC with equal-power transmitters (Section 3.2.1), we can use

(3.42) to write

COL(H, P/σ2) = CMAC(h11, . . . ,h1N , . . . ,hK1, . . . ,hKN , P/σ2). (3.47)

By grouping together K sets of N SIMO channels so that the MIMO channel

for user k is Hk, we get a new MAC that consists of K users each with N

antennas and power P/K. This ((K,N),M) MAC could achieve a higher

sum-rate capacity because the transmissions across the N antennas for each

user are now coordinated:

CMAC(h11, . . . ,h1N , . . . ,hK1, . . . ,hKN , P/σ2) ≤ CMAC({Hk} , P/σ2

).

(3.48)

From the MAC-BC duality (3.32), the BC sum-rate capacity is greater than

or equal to the dual MAC sum-rate capacity for a given power partitioning.

Therefore

CMAC({Hk} , P/σ2

) ≤ CBC({

HHk

}, P/σ2

). (3.49)

Given a (M, (K,N)) BC with channels HH1 , . . . ,HH

K , an upper bound on the

sum rate can be obtained by coordinating the detection across the K users.

This upper bound can be characterized by the closed-loop MIMO capacity

of the channel HH defined in (3.41):

CBC({

HHk

}, P/σ2

) ≤ CCL(HH , P/σ2). (3.50)

Because CCL(HH , P/σ2) = CCL(H, P/σ2), we can combine the results from

with (3.47),(3.48),(3.49), and (3.50) to obtain a chain of inequalities:

COL(H, P/σ2) ≤ CMAC({Hk} , P/σ2

)


≤ CBC({

HHk

}, P/σ2

)≤ CCL(H, P/σ2). (3.51)

For the special case of N = 1, the OL-MIMO capacity and MAC sum-rate

capacity are equivalent, resulting in the following:

COL(H, P/σ2) = CMAC({hk} , P/σ2

)≤ CBC

({hHk

}, P/σ2

)≤ CCL(H, P/σ2). (3.52)

Recall that at asymptotically high SNR, the open- and closed-loop ca-

pacities for a SU-MIMO channel are equivalent because CSIT does not pro-

vide any performance benefit (Section 2.2.2.2). It follows from (3.51) that

at asymptotically high SNR, the MIMO MAC and MIMO BC sum-rate ca-

pacities are also equivalent. Therefore for a fixed (and high) SNR P/σ2, the

sum-rate capacity does not depend on the direction of the transmission. Fig-

ure 1.9 shows the convergence of the CL-MIMO, MAC, and BC capacities at

high SNR. On the other hand, at low SNRs, CSIT for the SU-MIMO channel

provides an advantage, and there can be a significant difference in the MAC

and BC sum-rate capacity performance, as seen in Figure 1.10.

3.6.2 Sum rate versus SNR

By studying the sum-rate capacity performance of MU-MIMO channels more

carefully for low and high SNRs, we can gain insights into transmit strategies

for, respectively, low and high user geometries in cellular networks.

3.6.2.1 MAC sum rate at low SNR

We first consider the sum-rate capacity of the equal-power MAC (3.38) for

the case of asymptotically low SNR. Following the derivation for the open-

loop MIMO capacity for low SNR in Section 2.2.2.1, we have from (3.9) and

(3.38):

limP/σ2→0

CMAC({Hk} , P/σ2

) ≈ maxtrQk≤P/(σ2K)

tr

(K∑

k=1

HkQkHHk

)log2 e


=

K∑k=1

maxtrQk≤P/(σ2K)

tr(HkQkH

Hk

)log2 e

=

K∑k=1

P

σ2Kλ2max (Hk) log2 e, (3.53)

where λ2max (Hk) is the maximum eigenvalue ofHkH

Hk . As the SNR decreases,

the additive noise dominates the interference, and it becomes optimal for each

user to maximize its own rate without regard to its impact on the others [87].

It does so by transmitting with full power P/K on its dominant eigenmode

(3.53). If the channels of the users are statistically equivalent, the average

sum-rate capacity is K times the average rate of any user, and it follows that

limP/σ2→0

CMAC((K,N),M, P/σ2)

P/σ2= EHk

λ2max (Hk) log2 e, (3.54)

where EHkλ2max (Hk) is the expected value of the spectral radius taken over

realizations ofHk ∈ CM×N . The expected value can be computed analytically

for certain channel distributions including the i.i.d. Rayleigh distribution [88].

3.6.2.2 BC sum rate at low SNR

Recall that for the SU-MIMO channel at asymptotically low SNR, it be-

comes optimal to transmit a single stream over the best eigenmode if CSIT is

known. It follows intuitively that for the BC at low SNR, multiplexing data

for multiple users does not provide any advantage over single-user transmis-

sion. Indeed, at asymptotically low SNR, the BC capacity region converges

to the TDMA achievable rate region [89]. In this regime, the BC sum-rate

capacity and maximum TDMA sum rate are equivalent [90] and are achieved

by transmitting power P over the best eigenmode among the K users:

P/σ2 maxk

[λ2max

(HH

k

)]log2 e. (3.55)

In the limit of very low SNR, the ratio of the average sum rate to the SNR

can be expressed as

limP/σ2→0

CBC(M, (K,N), P/σ2)

P/σ2= lim

P/σ2→0

CTDMA(M, (K,N), P/σ2)

P/σ2

= E{HHk }max

k

[λ2max

(HH

k

)]log2 e, (3.56)


where the expectation is taken with respect to the channels HH1 , . . . ,HH

K .

Because the BC strategy transmits to the best among K users, the average

sum-rate capacity (3.56) improves as K increases due to multiuser diversity.

In contrast, average MAC sum-rate capacity (3.54) is independent of K (if

each user has power P/K).

3.6.2.3 MAC and BC sum rates at high SNR

Shifting to the high SNR regime, we recall that the multiplexing gain for a

(M,N) SU-MIMO channel is min(M,N), whether or not CSIT is available

(Section 2.2.2.2). From the chain of inequalities (3.51), it follows that

limP/σ2→∞

CMAC((K,N),M, P/σ2)

log2(P/σ2)

= limP/σ2→∞

CBC(M, (K,N), P/σ2)

log2(P/σ2)

= min(KN,M). (3.57)

Hence the sum-rate capacity of the BC and MAC scales with the minimum of

the total number of transmit or receive antennas. If K ≥ M , then increasing

the number of terminal antennas N provides combining gain but does not

increase the multiplexing order.

Because TDMA serves only a single user, the multiplexing gain is equiva-

lent to a SU-MIMO channel’s:

limP/σ2→∞

CTDMA(M, (K,N), P )

log2(P/σ2)

= min(M,N). (3.58)

If the BC transmitter does not have CSIT, it may not be able to effectively

multiplex signals for multiple users. In particular, for i.i.d. Rayleigh chan-

nels, the BC without CSIT achieves the same multiplexing gain of TDMA

(3.58) [91], indicating the importance of CSIT for the BC. For the MAC, if

CSIT is not available, each user can transmit isotropically to achieve a max-

imum sum rate of COL(KN,M,P/σ2). Therefore the full multiplexing gain

min(KN,M) can be achieved for the MAC without CSIT.

3.6.2.4 Numerical results

We first consider the performance of a BC and dual MAC for K = 2

single-antenna users with fixed channel realizations h1 and h2. The total

power P is the same for the two cases, and it is split evenly for the MAC


users. Figure 3.11 shows the rate regions for high and low SNR and for two

channel realizations — one where the two channels are nearly orthogonal

(∣∣hH

1 h2

∣∣ � ‖h1‖2 ≈ ‖h1‖2) and another where the channels are nearly col-

inear (∣∣hH

1 h2

∣∣ ≈ ‖h1‖2 ≈ ‖h1‖2).• As a result of the MAC-BC duality, the MAC capacity region is contained

within the BC capacity region in all cases.

• At high SNR for orthogonal channels, interference for the MAC is mini-

mal and each user can achieve its maximum rate simultaneously without

interference cancellation: log2(1 + P ‖h1‖2) ≈ log2(1 + P ‖h1‖2). As a re-

sult, the MAC capacity region has a rectangular shape. For the BC, user

k (k = 1, 2) can achieve rate log2(1 + P ‖hk‖2) bps/Hz if served alone.

By serving both users simultaneously and splitting the power evenly, each

user can achieve log2(1+P/2 ‖hk‖2), which is only 1 bps/Hz less than the

time-multiplexed rate.

• At high SNR for nearly co-linear channels, the interference for the MAC

is significant, and interference cancellation provides minimal benefit. The

BC capacity region is similar to the TDMA rate region, so superposition

coding provides little benefit over time sharing.

• At high SNR regardless of the channel realizations, the MAC and BC

sum-rate capacities are similar because the bounding OL and CL MIMO

capacities (3.51) are similar.

• At low SNR, the MAC capacity region would have a rectangular shape

regardless of the channels because the interference is dominated by the

noise power. The BC capacity region is nearly flat and is very similar to

the TDMA rate region. The BC has a higher sum-rate capacity, and it is

achieved by transmitting with power P to user 1.

Figure 3.12 shows the average sum rate versus SNR for the MAC, BC,

and TDMA channels for K = M users, M = 1 and 4, and N = 4 user

antennas. Figure 3.13 shows the same but with N = 1. (These two figures

can be compared with the corresponding curves for SU-MIMO in Figures 2.5

and 2.6.)

• For M = 1, the MAC, BC and TDMA channels are equivalent regardless

of N . At high SNR, they all achieve a multiplexing gain of M .

• For M = 4 at low SNR, the BC and TDMA sum rates are equivalent

because the rate regions are equivalent. At high SNR, the BC and MAC

sum rates are equivalent.


• For M = 4 and N = 4, TDMA achieves the same multiplexing gain as the

BC and MAC. The slopes are the same, but there is an offset in the rate

as a result of the single-user restriction for TDMA.

• For M = 4 and N = 1, TDMA cannot achieve multiplexing gain, and its

slope is the same as the single-antenna M = 1 channels.

• For M = 4 at high SNR, additional antennas at the users do not im-

prove the multiplexing gain because it is limited by M : min(M,K) =

min(M, 4K) = 4. However, additional antennas provide higher rates

through combining gain. A 6 dB combining gain can be achieved on av-

erage by each user, resulting in about a 2 bps/Hz rate improvement. At

20 dB SNR and higher, the net improvement in sum rate in using N = 4

antennas per user with K = 4 users is indeed about 8 bps/Hz.

Figure 3.14 shows the ratio of MU-MIMO average sum rates (3.44) and

(3.45) versus the SU-MIMO average rate, for M = K = 4 and N = 1 and 4:

CMAC((4, N), 4, P/σ2)

COL(1, N, P/σ2)and

CBC(4, (4, N), P/σ2)

COL(1, N, P/σ2). (3.59)

(For N = 1, comparisons can be made with the SU-MIMO capacity gain in

Figure 2.7.) As we will see in Chapter 6, this ratio provides insight into the

system tradeoffs between a single MU-MIMO link with M = 4 antennas at

the base and 4 isolated SU-SIMO links (each with the same power as the

MU-MIMO link) with M = 1 antenna at the base. The former is superior if

the ratio is greater than 4.

• At low SNR, the MAC achieves combining gain compared to the SU chan-

nel, and the BC achieves both combining and multiuser diversity gain.

This diversity advantage can be significant for very low SNR. The marginal

gains of MU-MIMO compared to SU-SIMO are diminished if the mobile

has multiple antennas.

• At high SNR, MU-MIMO achieves a multiplexing gain of 4 whereas SU-

MIMO achieves a multiplexing gain of 1. Therefore at high SNR, the ratio

approaches 4.

Figure 3.15 shows the ratio of the average sum rate between BC (3.45)

and TDMA (3.46) for M = K = 4 and for N = 1 and 4:

CBC(4, (4, N), P/σ2)

CTDMA(4, (4, N), P/σ2). (3.60)


• At low SNR, the BC and TDMA sum-rate capacities are equivalent, and

the ratio approaches 1 regardless of N .

• At high SNR, the multiplexing gain of the BC and TDMA are given re-

spectively by (3.57) and (3.58), and the ratio is

min(KN,M)

min(M,N). (3.61)

• For N = 4, both TDMA and BC can achieve a multiplexing gain of 4 so

the ratio approaches 1. For N = 1, TDMA cannot achieve multiplexing

gain so the ratio approaches 4. Therefore the relative gains of the capacity-

achieving BC techniques versus simple TDMA are diminished at high SNR

as the number of user antennas increases.

If we apply these observations to a cellular system, one general conclusion

is that if users are located far from the base transmitter (and therefore have

low geometries), spatial multiplexing of the users provides minimal gains over

simple TDMA transmission. However, if the users are close to the transmitter

(and therefore have high geometries), spatial multiplexing offers a significant

advantage.

3.6.3 Sum rate versus the number of base antennas M

We consider the sum-rate capacity of the (M, (K, 1)) BC and the ((K, 1),M)

equal-power MAC in the limit of a large number of users and antennas.

Because of the equivalence between the (K,M) OL SU-MIMO channel and

the ((K, 1),M) MAC, in the limit of large K and M with K/M → α, the

MAC sum-rate capacity converges almost surely to

limM,K→∞K/M→α

CMAC({hk} , P/σ2

)M

= CAsym(α, P/σ2), (3.62)

where the constant on the right side is defined in (2.49). We have the equiv-

alent result for the BC [92]


CBC({

hHk

}, P/σ2

)M

= CAsym(α, P/σ2) (3.63)


HighSNR

0 100

10

(bps/Hz)

(bps

/Hz)

0 100

10

(bps/Hz)

(bps

/Hz)

0 10

1

(bps/Hz)

(bps

/Hz)

0 10

1

(bps/Hz)

(bps

/Hz)

BC capacity region boundary

MAC capacity region boundary

Nearly orthogonal channels Nearly co-linear channels

LowSNR

Fig. 3.11 MAC and BC capacity regions for K = 2 single-antenna users with fixed

channels and with an equal power split for the MAC. At high SNRs, the sum rates of the

MAC and BC are equivalent, and they are significantly higher than the TDMA sum rate.

At low SNRs, the BC and TDMA sum rates are equivalent, and are slightly higher than

the MAC sum rate.

implying that equal distribution of power among users on the dual MAC is

asymptotically optimal with respect to the sum-rate capacity scaling. For the

(M, (K, 1)) TDMA channel, spatial multiplexing cannot be achieved because

each user has only a single antenna. Therefore:


CTDMA({

hHk

}, P/σ2

)M

= 0. (3.64)

Figure 3.16 shows the average sum rate of the BC, MAC and TDMA

channels versus M for K = M , N = 1, and i.i.d. Rayleigh channels. For low


0 10 20 30 400

5

10

15

20

25

30

35

SNR (dB)

Ave

rage

sum

rate

(bps

/Hz)

-20 -15 -10 -5 00

0.5

1

1.5

2

2.5

3

SNR (dB)

Ave

rage

sum

rate

(bps

/Hz)

BC (M,(M,4))MAC ((M,4),M)TDMA (M,(M,4))

M = 4

M = 1

M = 4

M = 1

Low SNR High SNR

Fig. 3.12 Average sum rate versus SNR for M = K = 1 or 4 and for N = 4 antennas per

user.

0 10 20 30 400

5

10

15

20

25

30

35

SNR (dB)

Ave

rage

sum

rate

(bps

/Hz)

-20 -15 -10 -5 00

0.5

1

1.5

2

2.5

3

SNR (dB)

Ave

rage

sum

rate

(bps

/Hz)

M = 4

M = 1

M = 4

M = 1


Low SNR High SNR

Fig. 3.13 Average sum rate versus SNR for M = K = 1 or 4 and for N = 1 antennas per

user.


-20 -10 0 10 20 30 401

2

3

4

5

6

SNR (dB)

Rat

io o

f ave

rage

sum

rate

BC (4,(4,N)) vs. (1,N)MAC ((4,N),4) vs. (1,N)

N = 4

N = 1

Fig. 3.14 Ratio of MU-MIMO average sum rates versus SU-SIMO average rate as a

function of SNR, for M = K = 4.

-20 -10 0 10 20 30 401

1.5

2

2.5

3

BC vs. TDMA, (4,(4,1))

BC vs. TDMA, (4,(4,4))

SNR (dB)

Rat

io o

f ave

rage

sum

rate

Fig. 3.15 Ratio of BC and TDMA average sum rates as a function of SNR, for

M = K = 4. At low SNR, the ratio approaches 1, and at high SNR, it approaches

min(M,KN)/min(M,N).


SNR, the slope of the sum-rate capacity curves for the BC and MAC approach

CAsym(1, P/σ2) as M increases, but there is a gap between the two. TDMA is

near optimal for small M . As the SNR increases, the BC-MAC performance

gap vanishes, and the multiplexing advantage over TDMA is apparent for

small M .

M

Ave

rage

sum

rate

(bps

/Hz)

0 5 10 150

10

20

30

40

50

60

70

80

M

Ave

rage

sum

rate

(bps

/Hz)

0 5 10 150

0.5

1

1.5

2

2.5

3

SNR = -10dB


SNR = 20 dBSNR = -10 dB

Fig. 3.16 Average sum rate versus M , for K = M and N = 1. As M increases asymptot-

ically, the sum-rate slope for both BC and MAC approach C∗(P ). The slope for TDMA

approaches 0.

3.6.4 Sum rate versus the number of users K

In cellular networks, the number of usersK vying for service could potentially

be much larger than the number of base antennas M . If we assume M ≥ N ,

then in the limit of large K under the assumption of block-fading Rayleigh

channels, we have from [86] that

limK→∞

CBC({

HHk

}, P/σ2

)log logKN

= M, (3.65)


where the convergence is almost surely. This implies that the multiplexing

advantage due to the transmit antennas M dominates over the multiuser

diversity achieved with the KN degrees of freedom at the receiver.

The sum-rate scaling in (3.65) can be achieved by treating each N -

antenna user as N independent single-antenna users, choosing M random

orthonormal beams for transmission, and in each of these beams trans-

mitting to the user antenna with the highest SINR [93]. When M and

P are fixed and K is large, the maximum SINR in each beam can be

shown to be P log(KN) + O (log log(KN)). This leads to a sum rate of

M log(P log(KN))+MO (log log log(KN)). The proof that this is the fastest

scaling possible can be found in [86].

It was shown in [94] that for N = 1 and correlated base antennas, there is

degradation in the sum rate, but the sum-rate scaling is the same as for the

uncorrelated case in (3.65).

If the CSI is not known at the transmitter and the MIMO channels are

i.i.d. Rayleigh, then [91]

limK→∞

CBC({

HHk

}, P/σ2

)log logK

= 0, (3.66)

again highlighting the importance of CSI at the transmitter for MU-MIMO.

As K increases asymptotically, the TDMA sum rate scales linearly with

min(M,N) and double-logarithmically with respect to K due to multiuser

diversity [86]

limK→∞

CTDMA(({

HHk

}, P/σ2

)log logK

= min(M,N). (3.67)

Combining (3.65) and (3.67), we have that the gain of multiuser transmission

over the best single-user transmission is:

limK→∞

CBC({

HHk

}, P/σ2

)CTDMA

({HH

k

}, P/σ2

) =M

min(M,N). (3.68)

Therefore multiuser transmission maintains an advantage as long as the num-

ber of transmitter degrees of freedom M is larger than the number of degrees

of freedom per user N .

Figure 3.17 shows the sum rate of the (4, (K,N)) BC and TDMA versus

K, for N = 1 and 4. From (3.68), the ratio of the sum rates approaches 4 and

1, respectively for N = 1 and 4. For the range of K considered, the TDMA

3.7 Practical considerations 115

sum rate achieves a significant fraction of the sum-rate capacity except for

the case of high SNR and N = 1.

For the MAC, we consider a modified case where each user transmits with

power P and has a single antenna. From (3.33), the sum-rate capacity is

CMAC({hk} ,KP/σ2

)= log2 det

(IM +

P

σ2

K∑k=1

hkhHk

). (3.69)

Since the channels are i.i.d. Rayleigh, we have that

limK→∞

1

K

K∑k=1

hkhHk = IM (3.70)

almost surely, as a result of the law of large numbers. Therefore, for large K,

CMAC({hk} ,KP/σ2

) ≈ log2 det

(IM +

KP

σ2IM

)(3.71)

≈ M log2

(KP

σ2

). (3.72)

It follows that

limK→∞

CMAC({hk} ,KP/σ2

)logK

= M. (3.73)

For the BC with single-antenna users (N = 1) and fixed total transmit power,

recall from (3.65) that the sum rate scales as M log logK as the number of

users K goes to infinity. On the MAC, however, each user brings its own

power of P so that the total transmitted power increases linearly with K.

This leads to the faster scaling of sum rate as M logK when K → ∞.


In this chapter we have described the capacity-achieving strategies for the

MAC and BC. These techniques are based on successive interference cancel-

lation and dirty paper coding, respectively. We have so far assumed these

techniques operate ideally. However, in reality, there are many practical chal-

lenges in implementation which we describe now.


10 20 30 40 50 60 700

10

20

30

40

K

Ave

rage

sum

rate

(bps

/Hz)

10 20 30 40 50 60 700

0.5

1

1.5

2

K

Ave

rage

sum

rate

(bps

/Hz)

BC (4,(K,N))TDMA (4,(K,N))

SNR = 20 dBSNR = -10 dB

N = 4

N = 1

N = 4

N = 1

Fig. 3.17 Average sum rate versus K for M = 4. For asymptotically high K, the ratio of

sum rates between BC and TDMA approaches M/min(M,N).

MAC BC TDMA

Fig. 3.18 Summary of sum-rate capacity scaling.


3.7.1 Successive interference cancellation

The capacity-achieving strategy for the MIMO MAC is difficult to implement

in practice from both the transmitter and receiver perspectives. At the trans-

mitter, the optimal transmit covariances for the N > 1 case are difficult to

compute because they require knowledge of all users’ MIMO channels. One

could do this by estimating channels at the central receiver, computing the

covariances, and communicating them to the users via control channels. This

strategy becomes challenging if the channels vary in time.

As a result of imperfect channel estimation, interference cancellation will

not be ideal. Even if a stronger user is decoded perfectly, errors in the channel

estimates for that user will lead to imperfect cancellation and therefore resid-

ual interference for weaker users. This becomes especially significant when the

SNR difference between the two users is high, and the problem is worse with

more users. In principle, partial interference cancellation with an MMSE cri-

terion could be used to mitigate this effect [95].

-10 -5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

Weaker user's SNR (dB)

Wea

ker u

ser's

rate

(bps

/Hz)

Perfect SIC

1% SIC error

10% SIC error

Fig. 3.19 Effect of cancellation error using SIC when detecting K = 2 users. The stronger

user has an SNR of 15 dB higher. As a result of the SIC error, the performance of the

weaker user is degraded.

For example, if we assume a 23 dB SNR difference between just two users,

then even a 1% error in cancelling the stronger user’s signal will result in a

-3 dB cap on the SINR that the weaker user can achieve. In practice, this


will mean that users with very disparate SNRs cannot be multiplexed, and

this will also affect the spectral efficiency gain that successive interference

cancellation can achieve. For illustration, Figure 3.19 shows the weaker user’s

rates for perfect cancellation, 1% cancellation error, and 10% cancellation

error, assuming the weaker user’s SNR is between -10 dB and 10 dB and the

stronger user is always 15 dB higher. The main point to note is the ceiling

imposed on the weaker user’s rate, especially evident in the 10% case.

Another issue is the sample resolution in the receiver’s analog-to-digital

(A-to-D) converter in order to handle the dynamic range of the received

signal. This can be modeled quite simply using results from rate-distortion

theory. If the average power of the received signal (including noise) is P , and

the A-to-D uses R bits/sym for quantization, then the quantization error can

be modeled as an additional independent Gaussian noise of variance P×2−R.

Naturally, this added noise will affect the weaker user more than the stronger

user.

3.7.2 Dirty paper coding

Dirty paper coding is difficult to implement due to the need for highly com-

plex multidimensional vector quantization [96] [97] [98]. The time-varying

nature of wireless channels makes the implementation even more difficult.

Dirty paper coding with single-dimensional quantization can be implemented

using Tomlinson-Harashima precoding, but there are significant performance

losses for the low SINR regime of wireless communication. DPC also requires

ideal channel state information at the transmitter which could be difficult to

acquire in practice. This problem is discussed in Section 4.4 in the context of

suboptimal broadcast channel techniques which also require CSIT.

3.7.3 Spatial richness

We recall that in SU-MIMO channels, multiple antennas are required at both

the transmitter and receiver to achieve spatial multiplexing. However, if the

antennas at either the transmitter or receiver array are fully correlated, then

spatial multiplexing is not possible (Section 2.4). In a BC with M antennas

and K single-antenna users, spatial multiplexing can be achieved even if the

3.8 Summary 119

M transmit antennas are fully correlated. For example if the channel is line-

of-sight, then a user’s channel vector would be determined by its direction

relative to the array elements (2.79). If the K users are sufficiently separated

in the azimuthal dimension, their channel vectors would be linearly indepen-

dent, and it would be possible to achieve full multiplexing gain min(M,K).

Furthermore, if each user had N fully correlated antennas, then multiplex-

ing gain M could be achieved as long as the users are spatially separated

and K ≥ M . Even if the users’ antennas were uncorrelated, the multiplexing

gain would be the same: M = min(M,KN). Similar arguments can be made

regarding the multiplexing gain of the MAC with fully correlated receive

antennas.

3.8 Summary

In this chapter, we studied two canonical multiuser channels: the MIMO mul-

tiple access channel (MAC) and the MIMO broadcast channel (BC), both

with additive white Gaussian noise (AWGN). The MIMO MAC models sev-

eral users transmitting to a single base station (uplink), and the MIMO BC

a single base station transmitting to several users (downlink).

• We examined the capacity regions and capacity-achieving techniques for

these channels, assuming perfect CSI everywhere. On the MIMO MAC,

receiver-side linear MMSE beamforming combined with sequential decod-

ing and interference cancellation of user signals is capacity-achieving. On

the MIMO BC, transmitter-side linear precoding combined with sequential

encoding through dirty paper coding (DPC) is capacity-achieving.

• We stated an elegant and very useful duality result relating the MIMO

MAC and MIMO BC, which allows us to translate BC problems to equiv-

alent MAC problems. Briefly, the duality result states that the capacity

region of the MIMO BC equals that of a dual MIMO MAC in which the

channel matrices are all conjugate-transposed and the users are allowed to

share power (with the sum power being the same as on the BC).

• We described algorithms for maximizing a weighted sum rate on the MIMO

MAC as well as on the MIMO BC. On the MAC, the optimal rate vector

always corresponds to decoding the users sequentially, in the order of in-

creasing weight (so that the user with the highest weight does not see any

interference). In contrast, on the BC, the optimal rate vector corresponds


to encoding the users using dirty paper coding in the order of decreasing

weight (so that the user with the lowest weight sees no interference). The

algorithm for the BC exploits MAC-BC duality.

• Finally, we investigated the sum-rate performance achievable on both

channels, and its asymptotic behavior with respect to SNR, number of

antennas, and number of users. We also compared the sum-rate capacity

to the sum rate achievable with simple TDMA. These results are summa-

rized in Figure 3.18.

Chapter 4

Suboptimal Multiuser MIMOTechniques

The previous chapter described the capacity regions of multiuser MIMO chan-

nels and techniques for achieving capacity. Due to the complexity of these op-

timal techniques, we are motivated to consider suboptimal multiuser MIMO

techniques with lower complexity.

For the multiple-access channel, we describe suboptimal beamforming

strategies for mobiles with multiple antennas and alternatives to the capacity-

achieving MMSE-SIC detector. For the broadcast channel, in order to avoid

the complexity of implementing dirty paper coding, we describe linear pre-

coding techniques which require only conventional single-user channel cod-

ing. These suboptimal precoding techniques can be classified according to

the knowledge available at the transmitter: full channel state information,

partial channel state information, statistical information, and no explicit in-

formation.

In the last section, we describe a duality for the MAC and BC under the

assumptions of linear processing and single-antenna users. This duality is

similar to the one for the general MIMO MAC and BC capacity regions and

will be useful for the system performance evaluation in Chapter 6.

4.1 Suboptimal techniques for the multiple-access

channel

In certain regimes that are relevant for cellular networks, the optimal capacity-

achieving techniques for the MIMO MAC can be simplified. In particular,

we recall that in the regime of high SNR for i.i.d. Rayleigh channels, op-


1214

122 4 Suboptimal Multiuser MIMO Techniques

timal sum-capacity scaling proportional to min(M,KN) can be achieved

without CSIT by having all K users transmitting isotropically over N an-

tennas. If K ≥ M , then optimal scaling of order M could be achieved by

having each user transmit with just a single antenna. Therefore, while the

capacity-achieving requirements for the MIMO broadcast channel and MIMO

multiple-access channel are similar (both require ideal CSIT, linear precod-

ing, ideal CSIR, and MMSE-SIC detection at the receiver), achieving near-

optimal MIMO MAC capacity is generally regarded as simpler in practice

because the requirement for CSIT is relaxed.

Transmitting a single stream per user is also optimal for the case of a

large number of users, as we describe below. We also describe some lower

complexity alternatives to using MMSE-SIC detection.

4.1.1 Beamforming for the case of many users

In general, the computation of the optimal precoder for each user in the

MIMO MAC depends on the channels of all users, and a particular user

could transmit multiple data streams. However, for a fixed M and N , as

the number of users K grows without bound, transmitting a single stream

per user with beamforming is both sufficient and necessary for achieving the

sum-capacity [99]. In other words, if the number of users is large, multiple

antennas at the mobile transmitter should be used for concentrating power

in a single stream rather than for spatial multiplexing. Unfortunately, this

result does not indicate how the beamforming should be performed.

For the related problem of achieving a fixed rate R/K for K users with

minimum total transmit power, it was shown in [100] that a simple strategy of

transmitting on each user’s dominant eigenmode is optimal for fixed M and

N as K increases asymptotically. This strategy is myopic in the sense that

user k requires knowledge of only its channel Hk. The covariance for user k is

PkvkvHk , where Pk is the transmit power and vk is the dominant eigenvector

of Hk. The sum-capacity can be achieved using a matched-filter receiver fol-

lowed by successive interference cancellation, and the transmit power for each

user is determined iteratively given the transmitter and receiver strategies.

Based on the observations in [99] and [100] and recalling that full-power

transmission is optimal when using an MMSE-SIC receiver, we consider

4.1 Suboptimal techniques for the multiple-access channel 123

the sum-rate performance of transmitting full power on each user’s domi-

nant eigenmode and using an MMSE-SIC receiver. Recall that the capacity-

achieving transmit covariance for each user is in general dependent on all

users’ channels. Therefore this alternative is simpler not only because each

user transmits only a single stream but because each requires knowledge of

only its own channel. Figure 4.1 shows the sum-rate performance averaged

over i.i.d. channels, compared with the sum-capacity for N = 4 transmit an-

tennas per user, an SNR of 10 dB per user, and M = 4 receive antennas.

As K grows, the proposed strategy’s sum rate approaches the sum-capacity.

We conjecture that the strategy is asymptotically optimal, with the intuition

being that as the number of users increases, interference for any user appears

spatially white such that myopic transmission without channel prewhitening

should be optimal.

0 5 10 15 205

10

15

20

25

30

35

40

Ave

rage

sum

rate

(bps

/Hz)

OptimumDominant eigenmode

Fig. 4.1 Sum rate versus K for M = 4 and N = 4, average SNR per user is 10 dB. The

suboptimal beamforming strategy achieves a significant fraction of the optimal sum rate,

even for moderate K.


4.1.2 Alternatives for MMSE-SIC detection

The capacity-achieving MMSE-SIC detector is an example of multiuser de-

tection, which refers to the class of techniques for detecting data signals in

multiple-access channels. The fundamentals of multiuser detection are given

in [43], and more recent advances in the field are described in [101]. Ear-

lier results in multiuser detection were often developed in the context of

code division multiple-access (CDMA) channels. In contrast to the MIMO

multiple-access channel where users’ signals are modulated by the channel

Hk, the users’ signals in CDMA channels are modulated by different spread-

ing sequences known to the receiver. The received signal model for the MIMO

MAC (3.1) is applicable for CDMA channels, and multiuser detection tech-

niques developed for these channels can therefore often be applied to the

MIMO MAC.

If the channel state information at the MIMO MAC receiver is not reli-

able, successive interference cancellation may actually degrade performance,

as discussed in Section 3.7.1. In this case, the linear MMSE receiver (with-

out cancellation)(Section 2.3.1) is simpler alternative to the MMSE-SIC and

achieves optimal multiplexing gain in the high-SNR regime for the SU-MIMO

channel. Because of the equivalence between the open-loop (K,M) SU-MIMO

channel and the ((K, 1),M) MIMO MAC, it also achieves optimal sum-rate

scaling for the MAC. We can further extend this result to state that the

MMSE receiver achieves optimal sum-rate scaling for the ((K,N),M) MIMO

MAC if users transmit isotropically.

We note that if an MMSE receiver is used, full power transmission would

not necessarily be optimal for weighted sum-rate maximization, as is the case

for the MMSE-SIC. We can see this by considering a ((2, 1),M) MAC where

both users have high SNR with h1 = h2. In this case, the user with the higher

QoS weight should transmit and the other user should be silent. In general,

optimizing the transmit powers for the MMSE receiver to maximize the sum

rate is a non-convex optimization because of the interference. (In Section 4.3,

an algorithm is given for determining the minimum sum of transmitted pow-

ers required to meet a set of SINR requirements for all users in a ((K, 1),M)

MAC.)

If the noise power dominates over the interference power, then the MMSE-

SIC provides minimal benefit over a simple matched filter. This situation

applies to three of the quadrants in Figure 3.11 (high SNR with nearly or-

4.2 Suboptimal techniques for the broadcast channel 125

thogonal channels, and low SNR for both nearly orthogonal and highly corre-

lated channels). In the fourth quadrant (high SNR with correlated channels),

TDMA transmission achieves a significant fraction of the MAC capacity re-

gion and also full multiplexing gain for the SU-MIMO channel. Even in cases

with moderate SNR, TDMA transmission may provide better performance

than simultaneous transmission and an MMSE receiver if the channels are

highly correlated.

4.2 Suboptimal techniques for the broadcast channel

Figure 4.2 shows an overview of suboptimal techniques for the broadcast

channel. The top branch distinguishes between single-user TDMA transmis-

sion and spatially multiplexed transmission for multiple users. We recall from

Section 3.6 that for asymptotically low SNR, TDMA transmission is actually

optimal. At high SNR, TDMA achieves multiplexing gain of min(M,N). Be-

cause the multiplexing gain of DPC is min(M,KN), TDMA achieves optimal

scaling if the number of antennas at each mobile is large (N ≥ M). In gen-

eral, time multiplexing of single-user transmission (using transmit diversity

or spatial multiplexing) could be used as a suboptimal strategy. However,

when the SNR is not too low and when the number of antennas per mobile is

small (N < M), we can do significantly better than TDMA using multiuser

transmission.

Under the multiuser transmission branch, we will focus on linear precoding

shown in Figure 4.3. If user k = 1, ...,K transmits Nk independent data

streams, its information bits are channel encoded to create Nk-dimensional

data vectors uk ∈ CNk×1. The encoding is done independently for each user,

so linear precoding is less complex than DPC. For a given symbol period, the

transmitted signal s is a linear combination of the user’s coded data symbols:

s =

K∑k=1

Gkuk, (4.1)

where Gk ∈ CM×Nk is the precoding matrix for user k. In the case of a

rank-one covariance matrix (Nk = 1), linear precoding is often known as

beamforming. In this case, the transmitted signal is


Nonlinear precodingLinear precoding

Multiuser transmissionSingle-user transmission

Suboptimal broadcast channel techniques

CSITprecoding

CDITprecoding

Codebook precoding

Random precoding

Fig. 4.2 Overview of suboptimal broadcast channel techniques.

s =

K∑k=1

gkuk = Gu, (4.2)

where G := [g1 . . .gK ] and u = [u1 . . . uK ]T. Because a mobile can demod-

ulate at most N spatially multiplexed streams, single-antenna mobiles can

demodulate beamformed signals with Nk = 1, but not precoded signals with

Nk > 1. For simplicity we will focus on the special case of beamforming but

often use the more general “precoding” terminology.

Mux, coding,

modulation

Mux, coding,

modulation

�

Pre-coding

Δ

Pre-coding

Δ

lΣ

�

Fig. 4.3 A linear precoding architecture for suboptimal downlink MU-MIMO transmis-

sion. The data stream of user k = 1, ...,K is channel encoded, and the data symbols uk

are precoded with the matrix Gk.


Linear precoding techniques can be categorized into four classes depending

on the knowledge available at the transmitter and the nature of the precoding

matrices Gk (or beamforming vectors gk). These classes are described briefly

here and treated in more detail in the remainder of the chapter.

• CSIT precoding: Knowledge of the channel state information (CSI)

HH1 , . . . ,HH

K is assumed at the transmitter. It is used when transmitter

CSIT is highly reliable, for example in TDD systems. Obtaining CSIT is

addressed in Section 3.4.

• CDIT precoding: Knowledge of the channel distribution information

(CDI), from which channel realizationsHH1 , . . . ,HH

K are drawn, is assumed

at the transmitter. CDIT precoding would be used when it is not possible

to obtain reliable CSIT, for example in TDD systems with very fast fad-

ing channels or in FDD systems where information at the transmitter is

obtained through an uplink feedback channel.

• Codebook precoding: Precoding matrices Gk are drawn from a code-

book B which consists of a fixed set of matrices. This is a generalization

of the single-user precoding technique described in Section 2.6. Like CDIT

precoding, codebook precoding is used when it is not possible to obtain

reliable CSIT. It is often used in FDD systems and is implemented by

having a user transmit on an uplink feedback channel an index into the

codebook to indicate its preferred precoding matrix.

• Random precoding: Precoding vectors are formed randomly, typically

with no explicit knowledge of the channels at the transmitter and with

minimal feedback from the mobiles. Random precoding is useful in ana-

lyzing the performance of networks with a large number of users. However,

because quality of service requirements are difficult to meet, it is typically

not used in practice.

We will see that linear precoding techniques like CSIT precoding can

achieve a signficant fraction of the optimal sum-rate capacity if the num-

ber of users K is large relative to M and if an effective user- selection al-

gorithm is used. For cases where K is small and where the CSI is known at

the transmitter, nonlinear precoding techniques have been shown to provide

performance gains over linear techniques by modifying the user data symbols

nonlinearly to optimize the spatial characteristics of the transmitted signal.

For example, under vector perturbation [102,103] the transmitted signal can

be written as x = G(u + τv), where G is a regularized zero-forcing matrix

and u is the vector of coded symbols. The data vector is perturbed by a


factor τv, where τ is positive real number and v is a K-dimensional complex

vector. These quantities are designed so that the perturbed data is nearly

orthogonal to the eigenvectors associated with the large singular values of

G. Another nonlinear precoding technique is a generalization of Tomlinson-

Harashima precoding [97]. A comparison of linear and nonlinear precoding

techniques is given in [104].

In the following sections, we describe the four classes of linear precoding

techniques in more detail. While each technique has different assumptions

about the transmitter knowledge, each mobile receiver is assumed to have

ideal knowledge of its own channel for demodulation.

4.2.1 CSIT precoding

Let us consider an (M, (K, 1)) BC using linear beamforming. The received

signal by the kth user is

xk = hHk s+ nk, (4.3)

where the transmitted signal s is given by (4.2) and the corresponding SINR

is

Γk =|hH

k gk|2v2kσ2 +

∑j =k |hH

k gj |2v2j, (4.4)

where vk = E[|uk|2

](k = 1, . . . ,K) is the power allocated to the kth user.

The total transmit power is

trE[ssH ] = tr[(GHG)E[uuH ]

](4.5)

=∑k

||gk||2vk. (4.6)

We can define an optimization problem for determining the beamforming

vectors gk and user powers vk (k = 1, ...,K) to maximize the weighted sum

rate subject to a transmit power constraint P . Given the quality of service

weights qk and channels hk, the optimization problem is

maxgk,vk

K∑k=1

qk log2

(1 +

|hHk gk|2vk

σ2 +∑K

j=1,j =k |hHk gj |2vj

)(4.7)

subject to vk ≥ 0, k = 1, . . . ,K∑k

||gk||2vk ≤ P.


In general, this is a non-convex optimization that can be solved using a

branch-and-bound method [105]. Unfortunately, this technique is computa-

tionally very intensive and cannot be implemented in real-time. (The related

problem of determining the minimum transmit power to meet a set of rate

requirements is discussed below in Section 4.3.)

The optimization problem (4.7) can be simplified by imposing a zero-

forcing constraint such that each user k = 1, . . . ,K receives no interference

from the signals of the other users:

|hHk gj | = 0, j �= k. (4.8)

To meet this constraint, it is sufficient for K ≤ M and for the channels

hk, k = 1, ...,K to be linearly independent. In other words, defining H :=

[h1 . . .hK ] to be the M ×K matrix of conjugate MISO channels, we require

K ≤ M and the columns of H to be linearly independent. Then by defining

the M ×K precoding matrix to be

G := H(HHH)−1, (4.9)

the K-dimensional received signal vector y := [y1 . . . yK ]T is

y = HHx+ n (4.10)

= HHGu+ n (4.11)

= u+ n. (4.12)

The zero-forcing criterion is satisfied becauseHHG = IK . (One could balance

the effects of interference and thermal noise using a regularized zero-forcing

precoding, which is analogous to the MMSE receiver for the MAC [102].)

Under zero-forcing, the kth user’s signal is received with SNR vk/σ2, and its

achievable rate is simply log(1 + vk/σ2). Using (4.9), the weighted sum-rate

optimization (4.7) can be rewritten as:

maxvk

K∑k=1

qk log(1 +

vkσ2

)(4.13)

subject to vk ≥ 0, k = 1, . . . ,K∑k

[(HHH)−1

](k,k)

vk ≤ P,

where the power constraint follows from (4.9).


In the case where q1 = . . . = qk, the problem is equivalent to maximiz-

ing the sum rate over parallel Gaussian channels, and waterfilling [37] gives

the optimal power allocation across the channels. A straightforward general-

ization of the waterfilling algorithm allows us to solve (4.13) when the QoS

weights are different.

The optimization problem in (4.13) can be generalized in a number of ways

that are useful for cellular networks. User selection addresses the problem of

zero-forcing precoding when the number of users K exceeds the spatial de-

grees of freedom M . In practice, when antennas are powered by individual

amplifiers, a per-antenna power constraint is more meaningful than the sum-

power constraint we have assumed. We are also interested in zero-forcing-

based transmission strategies for the case of mobiles with multiple antennas,

enabling the possibility of multiplexing multiple streams for each user. These

three extensions are addressed in the remainder of the section.

User selection

In deriving the zero-forcing precoding weights in (4.9), we ensured the exis-

tence of (HHH)−1 by requiring K ≤ M . In practice, the number of users K

could exceed M for networks with a moderate density of users. In order to

generalize (4.13) for the case where K > M , we let U ⊆ {1, . . . ,K} denote

the subset of users during an interval that are candidates for service. Defining

|U| to be the cardinality of U , we restrict |U| ≤ M . For a given subset U , werenumber the user indices and denote H(U) := [h1 . . .h|U|

]to be the M×|U|

matrix of MISO channels. We can rewrite the optimization (4.13) for a given

U as follows:

maxvk

|U|∑k=1

qk log(1 +

vkσ2

)(4.14)

subject to vk ≥ 0, k = 1, . . . , |U|∑k

[(H(U)HH(U))−1

](k,k)

vk ≤ P.

Letting R(U) be the weighted sum-rate solution to (4.14), the overall user-

selection problem maximizes R(U) over all possible subsets U :

maxU⊆{1,...,K}

R(U). (4.15)

(We note that for maximizing the weighted sum rate using DPC, the two-

level optimization is not necessary because the numerical algorithm in Section


3.5.2 automatically sets the covariances of inactive users to zero.) One could

solve (4.15) by performing the optimization (4.14) over all possible subsets

U and choosing the best subset. Using such a brute-force search, we would

consider each of the K users individually, all pairs of users, all triplets of

users, and so on up to all M -tuplets of users. The total number of possible

subsets would bemin(M,K)∑

j=1

K!

(K − j)!j!. (4.16)

The number of possibilities could be quite large for even modest K. For

example, with M = 10 and K = 20, we would need to consider over 100,000

subsets.

A greedy user-selection algorithm [106] can be used as a simpler alternative

to the brute-force method. The algorithm successively adds users in a greedy

manner to a candidate set of users. It starts by selecting the user which would

achieve the highest weighted rate if the transmitter were to serve only this

user. This user index is placed in the candidate set U . The algorithm then

calculates the weighted sum rate of all K − 1 pairs that include this user.

If the maximum weighted sum rate among the pairs does not exceed the

weighted rate of the single user, we exit the algorithm with U containing the

single user. Otherwise, we let U be the winning pair and try adding another

user. The algorithm proceeds iteratively in this manner until we run out of

either spatial degrees of freedom or users.

Greedy user-selection algorithm

1. Initialization

Set n := 1, Tn := ∅, DONE := 0

2. While n ≤ min(M,K) and DONE == 0, do the following:

a. k∗ := argmaxk/∈TnR (Tn ∪ {k})

b. if R (Tn ∪ {k∗}) < R (Tn)

U := Tn

DONE := 1

else

Tn+1 := Tn ∪ {k∗}n := n+ 1

end


A related heuristic user-selection algorithm based on the user channel

correlations rather than the user rates was proposed in [107]. Defining

RZF(M, (K, 1), P/σ2) to be the average sum rate for zero-forcing using this

selection algorithm, it was shown that this strategy achieves optimal sum-rate

scaling (3.65) as K increases:

limK→∞

RZF(M, (K, 1), P/σ2)

log logK= M (4.17)

Per-antenna power constraints

So far we have assumed a sum-power constraint (SPC) that limits the total

power summed across the M antennas. In practice, each transmit antenna is

powered by a separate amplifier, and therefore a power constraint applied to

each antenna would be more relevant. Under a per-antenna power constraint

(PAPC), the transmit power of antenna m can be written as E[|sm|2] =∑K

k=1 |gm,k|2vk, where gm,k is the (m, k)th element of the precoding matrix

G. Hence under a power constraint Pm for the mth antenna, the optimization

(4.13) becomes:

maxvk

K∑k=1

qk log(1 +

vkσ2

)(4.18)

subject to vk ≥ 0, k = 1, . . . ,K∑k

|gm,k|2vk ≤ Pm, m = 1, . . . ,M.

Figure 4.4 shows the power feasibility region for aK = 2-user,M = 2-antenna

case. The two axes correspond to the power vk given to each of the users.

For the PAPC, the feasible region is bounded by the linear power constraints

imposed by the two antennas and by the positive axes. Each antenna is

assumed to have a power limit of P/2. The feasibility region for the SPC

P is bounded by the positive axes and a single linear power constraint. The

PAPC region is contained within the sum power region, and the boundaries

of the two regions intersect at the vertex of the PAPC region. The optimal

operating point is the point lying on the feasibility region boundary which is

tangent to the sum-rate contour curve with the highest value. This problem

is a convex optimization which can be solved using conventional numerical

optimization techniques [108].

Figure 4.5 shows the sum-rate performance for a system with M = 4

transmit antennas serving K = 4 or 20 users, each with N = 1 antenna.


We consider three variations of ZF: with a sum-power constraint (SPC) and

greedy selection, with a per-antenna power constraint (PAPC) and brute-

force selection, and with PAPC and greedy selection. We compare zero-forcing

performance with optimal sum-rate capacity achieved using DPC and with

the TDMA transmission strategy where the user with the highest closed-loop

capacity is served (Section 3.6).

At very low SNR, TDMA is optimal. Therefore for lower SNR, the gap

between the sum-rate capacity and TDMA sum rate is smaller. At asymptot-

ically high SNR, TDMA achieves a multiplexing gain of min(M,N) = 1 while

the optimal multiplexing gain of the broadcast channel is min(M,KN) = 4.

The spatial multiplexing advantage of the zero-forcing techniques is clear at

higher SNRs because the slope of all three ZF curves (for a given value of

K) is the same as the sum-capacity slope, while the TDMA slope is much

shallower.

In going from K = 4 to 20 users, the ZF performance improves because

it is more likely to find more favorable channel realizations as the number of

users increases. However, the slope of the curves at high SNR is not depen-

dent on K because the multiplexing gain is always 4. For K = 4 users under

a per-antenna power constraint, brute-force ZF provides a slight performance

advantage over greedy ZF. However, forK = 20, there is visually no difference

between the two, and we show only a single curve for the PAPC performance.

Zero-forcing for N > 1: BD and MET

If the receivers have multiple antennas N > 1, then the extra degrees of

freedom could be used to demodulate multiple data streams. Let us first

consider the case where each user demodulates Nk = N streams and N is

such that M = NK. The received signal by user k is

xk = HHk s+ nk = HH

k

K∑j=1

Gjuj + nk, (4.19)

where Gk ∈ CM×N and uk ∈ C

N are, respectively, the precoding matrix

and data vector for user k. A generalization of the ZF technique for this

case is known as block diagonalization (BD) [85, 109], where the precoding

matrices are designed so that [H1 . . .HK ]H[G1 . . .GK ] ∈ C

KN×KN has a

block-diagonal structure consisting of K blocks of size N ×N along the diag-

onal and zeros elsewhere. Each user receives N streams with no interference

from the other users.


Per-antenna power constraint

Sum-power constraint

Contour lines of equal sum rate

Fig. 4.4 The shaded regions show the allowable power allocations under a sum-power

constraint and a per-antenna power constraint for K = 2 users and M = 2 transmit

antennas. The operating points for maximizing the sum rate under PAPC and SPC are

highlighted.

Under BD, the precoding matrix for the kth user is the product of two

matrices: Gk = G(L)k G

(R)k . The left matrix G

(L)k is size M × N and lies in

the null space of the other users’ MIMO channels: HHj G

(L)k = 0(N×N), j �= k.

The right matrix G(R)k is size N ×N and is the right unitary matrix of the

singular value decomposition (SVD) of HHk G

(L)k . In other words, given the

decomposition

HHk G

(L)k = V

(L)k ΛkV

(R)Hk , (4.20)

we assign G(R)k := V

(R)k . User k uses V

(L)Hk ∈ C

N×N as the linear combiner

for its received signal. As a result of the block-diagonal criterion, the combiner

output given the received signal (4.19) is

V(L)Hk xk = V

(L)Hk HkG

(L)k G

(R)k uk +V

(L)Hk nk

= V(L)Hk V

(L)k ΛkV

(R)Hk V

(R)k uk +V

(L)Hk nk

= Λkuk +V(L)Hk nk.

Because V(L)k is a unitary matrix, the resulting noise vector is spatially white

since nk is spatially white. Waterfilling can be used to allocate power across


0 5 10 15 20SNR (dB)

0

5

10

15

20

25

30

35

Ave

rage

sum

rate

(bps

/Hz)

0 5 10 15 200

5

10

15

20

25

30

35

SNR (dB)

Ave

rage

sum

rate

(bps

/Hz)

CSIT PC, PAPC, Brute-force

DPCCSIT precoding, SPCCSIT precoding, PAPCTDMA

Fig. 4.5 Average sum rate versus SNR for M = 4, K = 4, 20, N = 1. The average sum-

rate capacity for the broadcast channel (achieved using DPC) is compared with CSIT pre-

coding (zero-forcing) and TDMA transmission. Three variations of zero-forcing are shown:

sum-power constraint (SPC), per-antenna power constraint (PAPC) with brute-force user

selection, and PAPC with greedy-user selection. As SNR increases, all three zero-forcing

sum-rate curves have the same slope as the sum-rate capacity, indicating optimal scaling

with respect to M . With only a single-antenna per user, single-user transmission with

TDMA cannot achieve multiplexing gain.

the N data streams of uk. In doing so, the kth user achieves the closed-loop

MIMO capacity for the equivalent single-user MIMO channel HHk G

(L)k . The

BD technique is a generalization of both ZF precoding to the case of multiple

receive antennas and closed-loop single-user MIMO to the case of multiple

users.

The basic BD technique that we have just described is designed for the

specific case of M = KN where we transmit exactly N streams to each user.

Even though the transmitter could send up to M streams, this strategy may

not maximize the resulting sum rate. For example, if the channels between

users are highly correlated, imposing a ZF constraint could be too restrictive,

and the sum-rate performance could be improved by sending fewer streams.

A generalization of BD known asmultiuser eigenmode transmission (MET)

[110] maximizes the weighted sum rate by distributing up to M streams

among the K users so that each user receives up to N streams. (In general,


each user can receive up to min(N,M) streams, but we will assume that N ≤M .) Unlike BD, the number of streams sent to each user is not necessarily

N . MET is based on the SVD of each user’s channel HHk , characterized by

its N singular values. The base transmits up to N streams to each user in

the direction of its eigenmodes, and it uses a zero-forcing criterion so that its

users experience no interference.

We first describe the precoding and receiver combining for maximizing the

MET weighted sum rate for a given set of active users and active eigenmodes.

We let U be the set of active users. Given the SVD of HHk , k ∈ U , we let

Ek ⊆ {1, . . . , N} be the set of indices for its active eigenmodes. If |Ek| is thenumber of active eigenmodes for the kth user, the total number of eigenmodes

must satisfy∑

k∈|U| |Ek| ≤ M . We let Λk be the |Ek| × |Ek| diagonal matrix

whose components are the singular values of HHk corresponding to Ek. We

let V(L)k and V

(R)k be the corresponding submatrices of the left and right

unitary SVD matrices. To simplify notation, we do not explicitly write out

the dependence of Λk, V(L)k , and V

(R)k on Ek. We have that

V(L)Hk (Ek)︸︷︷︸|Ek|×N

HHk︸︷︷︸

N×M

= Λk(Ek)︸︷︷︸|Ek|×|Ek|

V(R)Hk (Ek)︸︷︷︸|Ek|×M

. (4.21)

As in the BD formulation, the precoding matrix for user k can be decomposed

into a left and right component: Gk = G(L)k G

(R)k . The matrix G

(L)k for MET

spans the null space of V(L)Hj HH

j for j ∈ U , j �= k, and it is of size

M × (M −∑

j∈|U|,j =k

|Ej |).

Unlike the BD design whereG(L)k needed to be orthogonal to allN dimensions

of the other users’ MIMO channels, G(L)k for MET is orthogonal to the |Ej | ≤

N selected eigenmode dimensions of the other active users j.

Processing the received signal xk with the |Ek| ×N combiner V(L)Hk and

relying on the zero-forcing properties of G(L)k , the combiner output is

V(L)Hk yk = V

(L)Hk HH

k G(L)k G

(R)k uk + (4.22)∑

j∈U,j =k

V(L)Hk HH

k G(L)j G

(R)j uj +V

(L)Hk nk

= ΛkV(R)Hk G

(L)k G

(R)k uk +V

(L)Hk nk.


We decompose the |Ek| × (M −∑j∈|U|,j =k |Ej |) matrix ΛkV(R)Hk G

(L)k , and

we write the SVD as

ΛkV(R)Hk G

(L)k =

[V

(L)k

] [Λk 0

] [V

(R)Hk

], (4.23)

where Λk is the diagonal matrix of |Ek| singular values. Letting the ma-

trix G(R)Hk be the top |Ek| × (M − ∑j∈|U|,j =k |Ej |) submatrix of V

(R)Hk ,

post-multiplying the combiner output in (4.22) with V(L)k results in a |Ek|-

dimensional vector:

V(L)Hk V

(L)Hk yk = V

(L)Hk Λk(Ek)V(R)H

k G(L)k G

(R)k uk + n′

k

= V(L)Hk

[V

(L)k

] [Λk 0

] [V

(R)Hk

]G

(R)k uk + n′

k

= Λkuk + n′k,

where processed noise vector n′k := V

(L)Hk V

(L)Hk nk is spatially white.

The eigenmodes for all active users are decoupled, and the power can

be allocated using waterfilling to maximize the weighted sum rate. If we

denote vk,j := E[|uk,j |2

]to be the power allocated to the jth eigenmode

(j ∈ Ek) of user k (k ∈ U), the transmitted signal for user k is Gkuk, and the

power transmitted for user k is trGHk diag[vk,1 . . . vk,|Ek|]Gk. For a given set

of active users k ∈ U and their active eigenmodes Ek, the weighted sum-rate

optimization is

maxvk,l

∑k∈U

qk∑l∈Ek

log2

(1 +

λ2k,lvk,l

σ2

)(4.24)

subject to vk,l ≥ 0, k ∈ U , l ∈ Ek∑k∈S

trGHk diag[vk,1 . . . vk,|Ek|]Gk ≤ P.

Similar to the optimization for the original ZF technique (4.14), the optimiza-

tion in (4.24) can be solved using weighted waterfilling over the∑

k∈U |Ek|parallel channels. Letting R(U , {Ek}) be the solution to (4.24) for a given

active set U and active eigenmodes Ek for k ∈ U , the outer optimization that

maximizes the weighted sum rate over all combinations of eigenmodes and

active user sets is

maxU⊆{1,...,K},Ek,k∈U,

R(U , {Ek}). (4.25)

To solve this optimization problem, one could resort to a brute-force search

as was done for the ZF case.


Alternatively, a generalization of the the user selection algorithm described

for ZF could be used for MET, where the selection occurs over KN virtual

users corresponding to the KN eigenmodes. Under MET, at most M streams

are transmitted and distributed among the K users so that an active user

receives at most min(M,N) streams. All the eigenmodes could be assigned

to a single user, or they could be distributed among multiple users. There-

fore closed-loop SU-MIMO is a special case of MET, and the optimization in

(4.25) automatically chooses between single-user and multiuser MIMO trans-

mission on a frame-by-frame basis.

Single-stream MET

If multiple users are served under MET each with a single stream, it is in gen-

eral not necessary for each user to be served on its dominant eigenmode (i.e.,

the mode with the largest singular value). The reason is that the dominant

eigenmodes of the users could be unfavorably correlated. However, the like-

lihood of these unfavorable correlations decreases as the number of users K

increases. It is shown in [111] that transmitting on the dominant eigenmode

for M out of K users is indeed optimal in the sense that the resulting sum

rate and the sum-rate capacity approach zero as K increases. Therefore, in a

similar way that single-stream transmission is asymptotically optimal for the

MAC (Section 4.1) for large K, single-stream transmission (beamforming) is

also optimal for the BC.

The MET optimization in (4.25) can be modified for single-stream trans-

mission by restricting the candidate eigenmode set Ek for the kth user to

contain at most a single eigenmode. We call this technique MET-1. The

single eigenmode is not necessarily restricted to be the dominant one be-

cause for smaller K it is useful to allow this flexibility. Compared to MET,

MET-1 reduces complexity in three ways. It reduces the control informa-

tion required to notify an active user which eigenmode to demodulate, it

reduces the user/eigenmode selection complexity, and it reduces the receiver

complexity since multi-stream demodulation is not necessary.

Figure 4.6 shows the transmission options for various transceiver strategies

for a (4,(2,2)) broadcast channel. The eigenmodes of user 1 are labeled 1A

and 1B, and those for user 2 are 2A and 2B. Eigenmodes 1A and 2A are

the users’ respective dominant eigenmodes. Under single-user transmission,

the transmitter chooses among the following four options for maximizing

the weighted sum-rate metric: {1A},{2A},{1A,1B},{2A,2B}. Under BD, the

base activates all four eigenmodes. MET has the most options to consider.


{1a}{2a} {1a,1b} { 2a,2b} {1a,2a} {1a,2b} {1b,2a} {1b,2b}

{1a,1b,2a} {1a,1b,2b} {1a,2a,2b}{1b,2a,2b} {1a,1b,2a,2b}

{1a} {2a} {1a,1b} {2a,2b}

{1a}{2a} {1a,1b} {1a,2a} {1a,2b} {1b,2a} {1b,2b}

{1a,1b,2a,2b}

SU

MET

MET-1

BD

User 1

User 22a2b

1a1b

Eigenmodes

Fig. 4.6 Transmission options for a (4, (2, 1)) BC using linear precoding. User 1 has

eigenmodes 1A and 1B, and user 2 has eigenmodes 2A and 2B. Potential active eigenmode

sets are shown for different transceiver options. SU-MIMO serves either user 1 or 2. BD

transmits on all eigenmodes. MET is the most general and considers all possible eigenmode

combinations. MET-1 transmits at most one stream to each active user.

Because of the spatial interaction among eigenmodes, transmission on non-

dominant eigenmodes could be optimal. For example, if 1A and 2A are highly

correlated, the maximum sum rate could be achieved by activating 1B and 2B.

Under MET-1, the transmission options are reduced. If the number of users

is small, as in this example, the sum-rate performance of MET-1 could suffer

significantly. However, as K increases, the average sum-rate performance of

MET-1 approaches that of MET, as shown in Figure 4.7.

Figure 4.8 shows the sum-rate performance versus SNR for M = 4, K =

4, 20, andN = 4. The sum-rate capacity (achieved with DPC), MET, MET-1,

and best single-user MIMO transmission are considered. Compared to Figure

4.5, where the mobiles had only a single antenna, all techniques have the

same multiplexing order 4, resulting in the same slope for high SNR. For

K = 4, there is a significant performance gap between MET and MET-1,


0 10 20 30 40 50 60 700

5

10

15

20

25A

vera

ge s

um ra

te (b

ps/H

z)

DPCCSIT precodingCSIT precoding, 1 streamTDMA

Fig. 4.7 Sum-rate performance versus the number of users K, for M = 4, N = 4, SNR =

10 dB. CSIT precoding using MET achieves a significant fraction of the sum-capacity.

As the number of users increases, CSIT precoding restricted to one stream (MET-1) also

achieves a significant fraction. Even though SU-MIMO achieves the full multiplexing order

of 4, the MU-MIMO options achieve higher sum rate because of the greater flexibility in

distributing streams among users.

0 5 10 15 20SNR (dB)

0 5 10 15 20SNR (dB)

0

5

10

15

20

25

30

35

Ave

rage

sum

rate

(bps

/Hz)

0

5

10

15

20

25

30

35

Ave

rage

sum

rate

(bps

/Hz)

DPCCSIT precodingCSIT precoding, 1 streamTDMA

Fig. 4.8 Sum rate versus SNR for M = 4, K = 4, 20, N = 4. MET achieves a significant

fraction of sum-capacity. Even though all three techniques have the same spatial degrees of

freedom, MET performs better than SU-MIMO due to multiuser diversity when K > M .


indicating that single-stream transmission is far from ideal. For high SNR,

TDMA performance approaches that of MET-1. If there are more users (K =

20), the gap between MET and MET-1 is negligible over the range of SNRs,

and there is a significant advantage over TDMA.

4.2.2 CDIT precoding

CDIT precoding for the broadcast channel assumes that at the transmitter,

the distributions of the K users’ channels are known but the realizations of

the channels are not known. The problem is to determine the precoders and

power allocations for the K users to maximize the expected weighted sum

rate. From (4.7), the required optimization is

maxgk,vk

K∑k=1

qkE

[log2

(1 +

|hHk gk|2vk

σ2 +∑K

j=1,j =k |hHk gj |2vj

)](4.26)

subject to vk ≥ 0, k = 1, . . . ,K∑k

||gk||2vk ≤ P.

For the special case of a single-user (K = 1) MISO channel, the opti-

mal precoding vector g1 corresponds to the eigenvector of E[h1hH1 ], and full

power P is applied in this direction [112]. For the single-user MIMO chan-

nel, the optimal transmit covariance Q1 for maximizing the average capacity

can be decomposed as Q1 = VPVH , where the eigenvectors of Q are the

columns of the unitary matrix V and where the eigenvalues are given by the

diagonal entries of P = diag {p1, . . . , pM}. The eigenvectors of the capacity-

achieving transmit covariance equal those of E[H1HH1 ], and the eigenvalues

can be found through a numerical algorithm to satisfy a set of necessary and

sufficient conditions [112]. Because the parallel channels are not orthogonal,

the power allocation does not correspond to a waterfilling solution.

For the MISO BC with multiple K > 1 users, the general solution for

maximizing the expected weighted sum rate (4.26) is not known. Some ad

hoc techniques have been proposed, including a variation of zero-forcing based

on channel statistics [113] and a technique combining CDIT and knowledge

of the instantaneous channel norm [114].

For the special case of a line-of-sight channel, the users’ directions can be

determined from the CDIT. Therefore knowledge of the channel hk can be


obtained implicitly from 2.79 (modulo the phase offset) and CSIT precoding

could be implemented. If ZF precoding is deployed, up to min(M,K) users

could be served on non-interfering beams.

4.2.3 Codebook precoding

In Section 2.3.7, we discussed codebook (or limited feedback) precoding for

the single-user MIMO channel. The receiver estimates the MIMO channel

and feeds back B bits to the transmitter as an index to indicate its preferred

precoding matrix chosen from a codebook containing 2B precoding matrices

(codewords). For the multiuser case, the codebook is known by the K users,

and each user feeds back B bits to indicate its preferred codeword. A block

diagram for codebook precoding for multiuser MISO channels is shown in

Figure 4.9.

bits

bits

Fig. 4.9 Block diagram for multiuser codebook precoding. User k estimates its channel

hHk and feeds back B bits to indicate its perfered precoding vector. The codebook B consists

of 2B codeword matrices and is known by the transmitter and the user.

In designing codebooks for single-user transmission, we recall that code-

words are designed to closely match the realizations or eigenmodes of the

channel. This principle can be applied in the multiuser case. However, when

serving multiple users, the interaction of the precoding matrices between

users needs to be considered. While the transmitter could serve a user with


its requested codeword, it could also decide to use another codeword because

the requested one would create too much interference for the other users.

We consider a (M, (K, 1)) broadcast channel and suppose for now that we

restrict the precoding vector to one drawn from the codebook B consisting of

M -dimensional codewords each with unit norm. To maximize the sum rate,

the precoding vectors for the K users g1, . . . ,gK ∈ B and power allocations

P1, . . . , PK need to be determined:

max∑k Pk≤P

maxg1,...,gK∈B

K∑k=1

log2

(1 +

|hHk gk|2Pk

σ2 +∑K

j =k,j=1 |hHk gj |2Pj

). (4.27)

This is a non-convex optimization which is difficult to solve. By further re-

stricting the power allocation among users to be equal, the optimal assign-

ment of codewords could be obtained through an exhaustive search if∣∣hH

k gj

∣∣is known for all k and j.

Alternatively, if equal power allocation is assumed and if the kth user knew

in advance the precoding vectors of the other K−1 users gj , j = 1, ...,K, j �=k, it would be straightforward to determine its optimal precoding vector:

gOptk = argmax

g∈Blog2

(1 +

|hHk g|2P/K

σ2 +∑K

j =k,j=1 |hHk gj |2P/K

). (4.28)

If there are K precoding vectors in the codebook, a user could determine its

best precoding vector using (4.28) if it assumes that for any vector it picks,

the other K − 1 vectors will be active.

PU2RC

This last strategy is the basis of per-user unitary rate control (PU2RC),

in which the precoding vectors are the columns of 2B/M (assumed to be

an integer) unitary matrices of size M × M . During a given transmission

interval, the active codewords are restricted to come from the columns of the

same unitary matrix. Therefore for a given candidate vector g drawn from

the unitary matrix u (u = 1, . . . , 2B/M), the mobile knows that the other

M − 1 active vectors are the other columns of this unitary matrix. It can

therefore use (4.28) to choose the best among 2B codewords.

If the number of users K is large compared to the number of codewords

2B , this strategy works well because it will likely find M users to serve who

are nearly orthogonal. However, if the number of users is small, their desired

vectors may be sparsely distributed among the columns of the unitary ma-


trices, and the number of users selecting any particular matrix would be less

than M . As a consequence, for a fixed K, the PU2RC sum rate decreases as

the size of the codebook 2B increases [115]. This result is somewhat counter-

intuitive since one would expect performance to improve as the knowledge of

the channel becomes more refined with increasing B.

Line-of-sight channels

In line-of-sight channels for single-user transmission, an effective codebook

could consist of 2B codewords that generate directional beams based on MRT.

Under multiuser transmission, the beamforming vectors could be redesigned

to reduce the interbeam interference compared to the MRT beams. Returning

to the example in Section 2.3.7.1, suppose we want to serve a 120◦ sector by

creating four beams in directions 45◦, 75◦, 105◦, 135◦ using a linear array with

M elements in Figure 2.17. If we partition the azimuthal dimension into four

30◦ sectors, the ideal beam response would be unity within the desired range

and zero outside this range. For example, the beam pointing in the direction

θ∗ = 105◦ would ideally have a unity response in the range (105◦ ± 15◦) andzero elsewhere, as shown in Figure 4.10.

The beamforming design for a linear array in line-of-sight channels is essen-

tially equivalent to the design of finite-impulse response (FIR) digital filters,

where the angular domain in the former is analogous to the frequency do-

main of the latter [116]. Therefore filter design techniques for minimizing

the frequency response in the stopband region can be used for minimizing

the sidelobe levels. Figure 4.10 shows the directional response of a more so-

phisticated beamformer based on a Dolph-Chebyshev design criterion. It has

significantly lower sidelobe levels with only a slightly wider main-lobe re-

sponse compared to the MRT response used for single-user beamforming.

Codebook-based zero-forcing

In general, the transmitter is not restricted to use the codewords in the

codebook B. While the feedback from the users index these codewords, the

actual precoding matrix for a given user could be a function of the selected

codewords of all K users.

For example, for the case of single-antenna users, suppose each user se-

lects the beamforming vector according to the single-user criterion (2.77). If

the base treats the desired beamforming vector of each user as the Her-

mitian conjugate of its actual MISO channel, it could perform codebook-

based zero-forcing precoding as described in Section 4.2.1. For K ≤ M , we


0 50 100 150 200 250 300 350-50

-40

-30

-20

-10

0

(degrees)

IdealMRTDolph-Chebyshev

Fig. 4.10 The directional response for a beamforming vector g pointing in the direction

θ∗ = 105◦. The ideal beam pattern response has unity gain over the directional range

of interest and zero gain outside this range. The beamformer g based on maximal ratio

transmission is matched to the channel h(θ∗). A more sophisticated beamformer based on

a Dolph-Chebyshev design has lower sidelobes, resulting in less interbeam interference for

users lying outside the range of interest.

would define H := [g1 . . .gK ] and use the precoding matrix defined in (4.9):

G := H(HHH

)−1.

A performance analysis of codebook-based zero-forcing in i.i.d. channels

is given in [117] using random codebooks, where 2B codewords are randomly

placed on the surface of an M -dimensional hypersphere. It was shown that

as the average SNR of the users increases, the number of feedback bits B

must be increased linearly with the SNR (in decibels) in order to achieve the

full multiplexing gain in an (M,M, 1) broadcast channel. This is in contrast

to the single-user MIMO link where no feedback is necessary at any SNR to

achieve full multiplexing gain. By modifying the metric for determining the

codebook feedback, one can improve the sum-rate performance for a given

number of feedback bits B. For example in [118], the feedback is chosen so

the resulting SINR is robust against all possible beam assignments at the


base. Generalizations of codebook-based zero-forcing precoding to the case

of multiple antenna receivers N > 1 are described in [60,119,120].

4.2.4 Random precoding

Under random precoding, beams are formed randomly instead of being drawn

from a predetermined codebook. These techniques work well when the num-

ber of users K vying for service is large. We first consider the case where a

single beam is formed to serve a single user at a time. We then consider the

more general case where M random mutually orthogonal beams are formed

to serve M users.

Single-beam

The concept was first proposed for the purpose of achieving multiuser di-

versity by artificially inducing SNR fluctuations through random beamform-

ing [121]. During frame t, a base equipped with M antennas creates a random

unit-power M -dimensional vector g(t) drawn from an isotropic distribution.

Each user measures and feeds back its SNR

P

σ2|hH

k g(t)|2, (4.29)

where the channel hk is an i.i.d. complex Gaussian channel and does not vary

with t. The base uses proportional scheduling, serving the user that achieves

the maximum weighted rate where the weight is the reciprocal average rate:

k∗(t) = argmaxk

1

Rk(t)log2

(1 +

P

σ2|hH

k g(t)|2). (4.30)

Under proportional fair scheduling, the long-term average rate as t approaches

infinity can be shown to exist, and each user is served a fraction 1/K of the

frames. If the number of users K is very large, the average rate achieved by

user k is:

limK→∞

[K lim

t→∞ Rk(t)]= log2

(1 +

P

σ2||hk||2

). (4.31)

In other words, its average rate is the same as if the base pointed a beam

g(t) = hk∗(t)/||hk∗(t)|| directly at the user everytime it was scheduled. This

beam would have required full CSIT, but random opportunistic precoding

achieves the same performance using only SNR feedback. The key is that ran-


dom precoding is coupled with opportunistic scheduling according to (4.30).

This performance gain would clearly not be achieved with random schedul-

ing. With a large number of users, the randomly formed beam will be well-

matched to some user during every frame. Even if the MISO channels had

different statistics, this technique would still work as long as the random

beams had matching statistics.

For correlated channels, g(t) creates a directional beam with a random

direction, and instead of using a random sequence of beams, one could use

an orderly cyclic scanning of the desired sector with fixed angular steps [121].

The beam sequence could be modified based on waiting times of users, re-

sulting in improved throughput and packet delays for the case of finite user

populations [122]. A shortcoming of using only a single beam is that this

strategy does not achieve linear capacity scaling with M . To address this

problem, we consider a generalization of the technique for multiple beams.

Multiple beams

As described in [93], a transmitter with M antennas creates a set of M

random orthonormal beams g1, . . . ,gM drawn from an isotropic distribution.

User k measures its SINR over each of the M beams (assuming power P/M

per beam) and feeds back its best SINR along with the corresponding beam

index m = 1, . . . ,M :

b∗ = arg maxb∈{1...M}

PMσ2 |hH

k gb|21 +∑M

j=1,j =bP

Mσ2 |hHk gj |2

. (4.32)

(The feedback overhead can be reduced by requiring feedback from only users

whose best SINR exceeds a threshold which is independent of K [93].) The

base transmits on M beams, where each beam is directed to the user with

the highest SINR. The sum rate of opportunistic beamforming averaged over

the channel realizations is given by:

ROB(M, (K, 1), P/σ2) =

Eh1...hK

M∑b=1

log2

(1 + max

i∈{1...K}

PMσ2 |hH

i gb|21 +∑M

j=1,j =bP

Mσ2 |hHi gj |2

). (4.33)

As the number of users increases asymptotically, multi-beam opportunistic

precoding achieves the optimal sum-rate scaling:

limK→∞

ROB(M, (K, 1), P/σ2)

log logK= M. (4.34)


Intuitively, this strategy achieves linear scaling with M because it becomes

increasingly likely as K increases to find a set of M users whose channels are

orthogonal and match the M random beams. In fact, opportunistic precoding

not only achieves optimal sum-rate scaling but actually achieves optimal sum

rate asymptotically:

limK→∞

[ROB(M, (K, 1), P/σ2)− CBC(M, (K, 1), P/σ2)

]= 0, (4.35)

where the average sum rate for the broadcast channel is defined in (3.45).

Even though the sum rate scales optimally, there is a significant performance

gap as seen in Figure 4.11. Strategies for narrowing the performance gap for

sparse networks are described in [123].

Note that whereas the single-beam case considered scheduling over static

channels for each user, the multibeam sum rate in (4.33) considers the max-

imum “one-shot” sum rate averaged over random realizations for each user.

Under this formulation, the average sum rate does not rely on the orthonor-

mal beams to be random. Therefore codebook precoding with an arbitrary

fixed set of M orthonormal beams in the codebook achieves the same perfor-

mance and requires the same feedback (SINR and beam index for each user)

as random precoding.

In practice, we would be interested in the scheduled performance for a finite

number of users. We can compare codebook precoding using an arbitrary set

of M orthonormal beams that do not change in time with random precoding

using a set of beams that change during each frame. If for a given set of K

users the channels h1, . . . ,hK each change randomly from frame to frame

(with infinite doppler), then the underlying statistics of the SINR in (4.32)

do not depend on whether the beams g1, . . . ,gM are fixed in time or not.

The scheduled performance is therefore equivalent for the two strategies.

On the other hand, if the user channels are fixed over time (zero doppler),

then codebook precoding has a disadvantage because a user’s channel could

have a poor orientation with respect to the codebook vectors, and its mis-

fortune does not change with time. With random precoding, however, the

relationship of a user’s channel with the random vectors changes each frame,

allowing it to take advantage of favorable beam realizations. If hierarchical

feedback is allowed for codebook precoding, the precoding vectors could be

refined over time as described in Section 2.6, and its performance would sur-

pass that of random precoding.

4.3 MAC-BC duality for linear transceivers 149

0 10 20 30 40 50 60 700

5

10

15

20A

vera

ge s

um ra

te (b

ps/H

z)

DPC

CSIT precoding

Random precoding

Fig. 4.11 Mean sum rate versus number of users K. Average user SNR is SNR = 20 dB.

Random precoding requires no explicit CSI information at the transmitter and achieves the

sum-rate capacity asymptotically as K increases. However its performance is far from opti-

mal for reasonable K. On the other hand, ZF requires full CSIT but achieves a significant

fraction of the sum-rate capacity for all K.

4.3 MAC-BC duality for linear transceivers

In this section, we will describe a duality between the ((K, 1),M) MAC and

the (M, (K, 1)) BC that holds under the restriction of linear processing, i.e.,

when the MAC receiver and BC transmitter are constrained to implement

only linear beamforming at their M antennas, in order to separate the signals

among the K users. This means that successive interference cancellation is

not permitted on the MAC, and dirty paper coding is not permitted on the

BC. This duality is similar in spirit to the one described for the MAC and

BC capacity regions (Section 3.4), and it will be used for the simulation

methodology in Section 5.4.4.

For the purpose of stating the duality results, it will be convenient to

normalize the noise power at each antenna to 1 in the MAC and BC channel

models of (3.1) and (3.2). We will accordingly model the MAC by


x =

K∑k=1

hksk + n, (4.36)

where n ∼ CN (0, IM ), and the BC by

xk = hHk s+ nk, k = 1, 2, . . . ,K, (4.37)

where nk ∼ CN (0, 1) for each k.

On the MAC, suppose user k transmits at power pk and is received with

the unit-norm receiver beamforming vector wk. The resulting SINR for user

k is then

αk =

∣∣wHk hk

∣∣2 pk1 +∑

j =k

∣∣wHk hj

∣∣2 pj . (4.38)

Denote by ΓMAC the set of all such achievable SINR vectors (α1, α2, . . . , αK),

over all choices of the beamforming vectors w1,w2, . . . ,wK and powers

p1, p2, . . . , pK .

Turning our attention now to the BC, suppose the base station uses power

qk and the unit-norm transmitter beamforming vector wk for user k. Then

the resulting SINR for user k is

βk =

∣∣hHk wk

∣∣2 qk1 +∑

j =k

∣∣hHk wj

∣∣2 qj . (4.39)

Denote by ΓBC the set of all such achievable SINR vectors (β1, β2, . . . , βK),

over all choices of the beamforming vectors w1,w2, . . . ,wK and powers

q1, q2, . . . , qK .

The duality result for linear processing, originally due to [124] and [125],

states that the achievable SINR regions ΓMAC and ΓBC on the MAC and

BC, respectively, are identical:

ΓMAC = ΓBC . (4.40)

Further, any feasible SINR vector (γ1, γ2, . . . , γK) can be achieved with the

same beamforming vectors and the same sum power in both channels, i.e.,

there must exist unit-norm beamforming vectors w1,w2, . . . ,wK , MAC pow-

ers p1, p2, . . . , pK , and BC powers q1, q2, . . . , qK for the K users, such that∣∣wHk hk

∣∣2 pk1 +∑

j =k

∣∣wHk hj

∣∣2 pj = γk =

∣∣hHk wk

∣∣2 qk1 +∑

j =k

∣∣hHk wj

∣∣2 qj (4.41)

4.3 MAC-BC duality for linear transceivers 151

andK∑

k=1

pk =

K∑k=1

qk. (4.42)

4.3.1 MAC power control problem

On the MAC, given feasible SINR targets γ1, γ2, . . . , γK for the K users, we

wish to find transmitted powers p1, p2, . . . , pK and unit-norm receiver beam-

forming vectors w1,w2, . . . ,wK that minimize the sum of the transmitted

powers while meeting all the SINR targets. More formally, we have the opti-

mization problem

min{pk},{wk}

K∑k=1

pk (4.43)

subject to ∣∣wHk hk

∣∣2 pk1 +∑

j =k

∣∣wHk hj

∣∣2 pj ≥ γk, ‖wk‖ = 1, pk ≥ 0 ∀k. (4.44)

The following simple iteration has been shown to solve the above problem

[126], regardless of how the user powers are initialized:

MAC power control algorithm

1. Given powers p(n)1 , p

(n)2 , . . . , p

(n)K for the K users, let w

(n)k be the unit-

norm vector that maximizes user k’s SINR:

w(n)k =

Q(n)k hk∥∥∥Q(n)k hk

∥∥∥ , where Q(n)k =

⎛⎝IM +

∑j =k

p(n)j hjh

Hj

⎞⎠−1

.

(4.45)

2. Update user k’s power to just attain its target SINR of γk, assuming

that every other user j continues to transmit at power p(n)j and user

k’s receiver beamforming vector is w(n)k :

p(n+1)k =

γk

(1 +∑

j =k

∣∣∣(w(n)k )Hhj

∣∣∣2 p(n)j

)∣∣∣(w(n)

k )Hhk

∣∣∣2 . (4.46)


4.3.2 BC power control problem

On the BC, given feasible SINR targets γ1, γ2, . . . , γK for theK users, we wish

to find transmitted powers q1, q2, . . . , qK and unit-norm transmitter beam-

forming vectors w1,w2, . . . ,wK that minimize the sum of the transmitted

powers while meeting all the SINR targets. More formally, we have the opti-

mization problem

min{qk},{wk}

K∑k=1

qk (4.47)

subject to ∣∣hHk wk

∣∣2 qk1 +∑

j =k

∣∣hHk wj

∣∣2 qj ≥ γk, ‖wk‖ = 1, qk ≥ 0 ∀k. (4.48)

The following simple iteration has been shown to solve the above problem

[124], regardless of how the user powers are initialized:

BC power control algorithm

1. Given downlink powers q(n)1 , q

(n)2 , . . . , q

(n)K and virtual uplink powers

p(n)1 , p

(n)2 , . . . , p

(n)K for the K users, let w

(n)k be the unit-norm vector

that maximizes user k’s SINR on the virtual uplink:

w(n)k =

Q(n)k hk∥∥∥Q(n)k hk

∥∥∥ , where Q(n)k =

⎛⎝IM +

∑j =k

p(n)j hjh

Hj

⎞⎠−1

.

(4.49)

2. Update user k’s downlink and virtual uplink powers to just attain

its target SINR of γk on both the downlink and the virtual uplink,

assuming that the powers for every other user j remain at q(n)j (down-

link) and p(n)j (virtual uplink), and user k’s beamforming vector is

w(n)k on both the downlink and the virtual uplink:

p(n+1)k =

γk

(1 +∑

j =k

∣∣∣(w(n)k )Hhj

∣∣∣2 p(n)j

)∣∣∣(w(n)

k )Hhk

∣∣∣2 , (4.50)

q(n+1)k =

γk

(1 +∑

j =k

∣∣∣hHk w

(n)j

∣∣∣2 q(n)j

)∣∣∣hH

k w(n)k

∣∣∣2 . (4.51)



In this section, we describe challenges for implementing downlink precoding

in practice, including the acquisition of CSIT and the auxiliary signalling

such as pilot signals.

4.4.1 Obtaining CSIT in FDD and TDD systems

CSIT precoding requires ideal knowledge of the users’ channel state informa-

tion at the transmitter. In frequency-division duplexed (FDD) systems, the

downlink and uplink transmissions occur on different bandwidth resources.

Mobiles could estimate the CSI based on downlink pilot signals, quantize the

estimate, and feed back this information to the base over an uplink channel.

This is a special case of feedback for codebook precoding where the code-

words designed to approximate the channel realizations. The quality of the

CSI used by the transmitter would depend on the quality of the initial down-

link estimate, the bandwidth of the uplink feedback channel, and the time

variation of the channel.

Alternatively, one could use unquantized analog feedback by modulating

an uplink control signal with the estimated channel coefficient. An analysis of

zero-forcing beamforming with realistic channel estimation and both quan-

tized and unquantized feedback is given in [127]. Its conclusion is that very

significant downlink throughput is achievable with simple and efficient chan-

nel state feedback, provided that the feedback link over the uplink MAC is

properly designed. A comprehensive survey of techniques for communicating

under limited feedback is found in [128].

In time-division duplexed (TDD) systems, transmissions for the uplink and

downlink are multiplexed in time over the same bandwidth. By estimating

the user channels on the uplink (based on user pilot signals), the base could

use these estimates for the following downlink transmission. If the users were

transmitting data on the uplink, their channels would need to be estimated

for coherent detection. Therefore by exploiting the channel reciprocity, the

base could obtain CSI for downlink transmission with no additional overhead.

However, because the uplink CSI estimates are based on uplink pilot channels,

the pilot overhead increases as the number of usersK and number of antennas

per user N increases. For a fixed overhead, the reliability of the CSIT depends


on the duplexing interval relative to the channel coherence time. Overall,

ideal CSIT could be a reasonable assumption for a TDD system with a small

number of slowly moving users.

In practice, calibration of the RF chains is also required to ensure channel

reciprocity [129]. This procedure can be performed within a fraction of a

second and relatively infrequently as the electronics drift (on the order of

minutes), so it does not contribute significantly to the overhead.

While TDD-enabled CSIT results in better performance than an FDD

system with limited feedback, the system implementation of TDD is often

more restrictive because the downlink and uplink transmissions among all

bases in the network must be time synchronized in order to avoid potentially

catastrophic interference. In other words, all bases must transmit at the same

time and receive at the same time. As seen in Figure 4.13 if the two bases are

not time-synchronized, a downlink transmission by base A could cause severe

interference for the uplink reception of base B. The interference power from

base A could be significantly higher than the desired user’s signal because

the base station transmit power is much higher than the mobile power and

because the channel between the two bases is potentially unobstructed. The

received interference power could be high enough to overload the sensitive

low-noise amplifiers at the front end of base B and to prevent any baseband

detection or interference mitigation techniques from being used.

This synchronization requirement applies to base stations owned by dif-

ferent operators that share the same tower. Even though the different bases

operate on different frequencies, the out-of-band interference could still be

detrimental if the base stations are not synchronized. Therefore while TDD

systems offer the potential of adjusting the duplexing fraction between uplink

and downlink transmissions, this adjustment must be made on a network-

wide basis in order to prevent severe interference.

4.4.2 Reference signals for channel estimation

Pilot signals (or reference signals as they are known in the 3GPP standards)

are sent from the transmitter and used to enable CSI estimation at the re-

ceiver for coherent demodulation. There are two classes of reference signal

structures, as shown in Figure 4.14.


Uplink Downlink

Fig. 4.12 For TDD systems, channel reciprocity is used to obtain CSIT. On the uplink,

pilots are transmitted by the users, and the base estimates the channels H1, . . . ,HK ∈CN×M . On the downlink, the base uses the complex conjugate of each estimate for trans-

mission.

A BFig. 4.13 Under TDD, if base stations A and B are not time-synchronized, a downlink

transmission by base A will cause severe interference for the uplink reception of base B.

Using common reference signals, a reference signal pm ∈ CT spanning T

time symbols is transmitted on antenna m = 1, ...,M . The duration T is de-

signed so that the channel is relatively static over these symbols. (In general,

the reference signals can span both the time and frequency domains, for ex-

ample, in OFDMA systems.) The reference signals are mutually orthogonal

in the time dimension: pHmpn = 0 for any m �= n. Therefore the length of

the reference signals must be at least as large as the number of antennas:

T ≥ M . Mobile k correlates the received signal with each of the known ref-

erence signals to obtain an estimate of the channel hk. The reference signals

are “common” in the sense that they are utilized by all users.

Using dedicated reference signals, a reference signal is assigned to each

beam. Figure 4.14 shows an example where dedicated reference signals trans-

mitted for 2B codewords in codebook B. By correlating the received signal

by each of the reference signals p1, ...,p2B , the mobile obtains estimates of

the beamformed channels hHk g1, ...,h

Hk g2B . Dedicated reference signals are

sometimes known as user-specific reference signals when a beam is assigned

to a specific user. In this case, the assigned user correlates the received signal

with only the associated reference signal assigned to it.


Dedicated reference signals benefit from the transmitter combining gain

of the beamforming vector, resulting in a greater range for a fixed reference

signal power compared to the common reference signals. For coherent de-

modulation of a signal modulated with precoding vector g, user k requires

an estimate of the equivalent channel hHk g. Dedicated reference signals must

be used when the precoding vector is not known at the mobile, for exam-

ple with CSIT or random precoding. Otherwise, if the precoding vector g is

known, it could first estimate the channel hHk using the common reference

signals and then compute the estimate for hHk g.

M

Common reference signals

Dedicated reference signals

ΣM M

Fig. 4.14 Two types of reference (pilot) symbols. Common reference symbols allow mobile

k to estimate its channel hHk . Dedicated reference symbols allow mobile k to estimate the

beamformed channels hHk g1, . . . ,hH

k g2B .

4.5 Summary

This chapter describes techniques for the MIMO MAC and BC channels

that provide suboptimal performance but with lower complexity compared

to capacity-achieving techniques.

• For the MAC, the design of optimal transmit covariances for users with

multiple antennas is difficult. Beamforming (transmitting with a rank-one

covariance) becomes optimal as the number of users increases asymptoti-

cally.

4.5 Summary 157

• For the BC, we focus on precoding techniques where linear transformations

are applied to independently encoded user data streams. CSIT precoding

is best suited to TDD systems where CSI is readily available at the base

station transmitter through channel reciprocity. Zero-forcing beamforming

is a type of CSIT precoding in which active users receive their desired data

signal with no interference. CDIT and codebook precoding are suited to

FDD systems where accurate CSIT is not available. Random precoding is

suitable for systems with a large number of users.

• For the single-antenna MAC and BC constrained to linear processing, we

also describe simple iterative algorithms for minimizing the sum power

transmitted while attaining given feasible SINR targets for all the users.

The algorithm for the MAC actually minimizes every user’s power indi-

vidually.

Chapter 5

Cellular Networks: System Model andMethodology

In the previous chapters we discussed MIMO techniques for isolated single-

user and multiuser channels where the received signal is corrupted only by

thermal noise. In this and the following chapter, we study MIMO techniques

in cellular networks where the performance is corrupted by co-channel inter-

ference in addition to thermal noise. In this chapter, we describe a cellular

network model and methodology for evaluating its performance by numerical

simulation. The following chapter presents performance results and insights

for MIMO system design.

Detailed performance evaluation of MIMO techniques in cellular networks

is challenging due to the complexity of these networks. One would have to

model each component of the system in great detail, accounting for practical

attributes and impairments such as user mobility, spatial correlation, finite

data buffers, channel estimation errors, hybrid ARQ, standards-dependent

frame structure, data decoding, and errors in the control channels. Modeling

these details in a simulation would result in accurate performance predictions,

at a cost of great complexity and loss of generality.

At the other extreme, purely analytic techniques have been applied to a

simplified cellular network model known as the Wyner model [130] where

bases and users lie on a line instead of a plane. This one-dimensional model

provides many fundamental insights of theoretical interest. However, their

applicability to more practical cellular networks is limited because the model

does not account for basic attributes such as shadowing and random user

placement.

Hybrid semi-analytic techniques have also been proposed to bridge the

gap between analysis and detailed simulations [131–134]. These techniques


1595

160 5 Cellular Networks: System Model and Methodology

have wider applicability than the purely analytic ones, but they have limited

flexibility because they apply only to specific antenna configurations.

The performance evaluation technique used in this book is based on Monte

Carlo simulations, which capture key aspects of a generic cellular network. In

order to focus on fundamental insights, we use performance metrics based on

Shannon capacity described in earlier chapters and make simplifying assump-

tions which allow us to understand the relative performance of the various

MIMO techniques. The methodology can be applied to different antenna

architectures and accounts for system elements that are common to contem-

porary cellular networks including IEEE 802.16e, 802.16m, LTE, and LTE

Advanced.

Section 5.1 provides a broader overview of the cellular network design

problem described in Chapter 1. Section 5.2 describes the general system

model for performance evaluation, including the model for sectorization, the

reference SNR parameter, and a technique for accounting for interference at

the edge of the network. In Section 5.3, two modes of base station opera-

tion are discussed along with various forms of spatial interference mitigation.

Section 5.4 describes the performance metrics and the simulation methodol-

ogy for the numerical simulation results in Chapter 6. Section 5.5 describes

implementation aspects of contemporary packet-scheduled cellular systems,

including sectorization, scheduling, and acquiring channel state information

at the transmitter (CSIT).

Depending on the context, a cell site (or simply site) is defined to be

either the hardware assets associated with a hexagonal cell or the hexagonal

cell itself. Each cell site is partitioned into S sectors, and, unless otherwise

noted, a base station with M antennas is associated with each sector.

5.1 Cellular network design components

A framework for cellular network design can be given as follows: for a given

geographic area defined by the characteristics of its channels and expected

data traffic, design the medium access control (MAC) and physical (PHY)

layers, a base station architecture, a placement of base stations, and a ter-

minal architecture to optimize the spectral efficiency per unit area, subject

to resource and cost constraints. (From the context, it will be clear whether

“MAC” stands for “medium access control” or “multiple-access channel”.)

5.1 Cellular network design components 161

This framework is illustrated in Figure 5.1, and we consider each of the com-

ponents below.

Channel statisticsPathlossAngle spreadDelay spreadDoppler spreadShadowing

User statisticsNumber of usersLocationMobilityData characteristics

Fundamental constraintsPowerBandwidthCarrier frequencyDuplexing

Cost constraintsOperation expensesCapital expenses

MAC / PHY layersResource allocationMultiple access Frame structureControl/data signalingMIMO techniques

Base station designBase locationsSectorization orderBackhaul/coordinationProcessing capabilityAntenna array architecture

Terminal designProcessing capabilityAntenna array architecture

Radio access cellular network

design

Fig. 5.1 Overview of the characteristics, constraints, and outputs for a radio access cellular

network design.

Channel characteristics

The system should be designed to match the statistics of the channel, which

depend on factors such as the terrain (urban, suburban, rural), the location

of the users (indoors or outdoors), and the user characteristics. The channel

is characterized by its distance-based pathloss, its shadowing characteristics,

and its variations in time, frequency and space [39]. For the distance-based

pathloss, we use a simple slope-intercept model [135] to describe the signal

attenuation as a function of the distance traveled relative to the pathloss

at a fixed distance given by the intercept value which is dependent on the

carrier frequency. Whereas in free space the power decays with the square

of distance, signals in most cellular channels decay with a coefficient closer

to the model of an ideal reflecting ground plane whose value is 4. Channel

modeling will be discussed in Section 5.2.


Variations in the time, frequency, and spatial domains are measured re-

spectively by the doppler spread, delay spread, and angle spread. These pa-

rameters are dependent on the carrier frequency. In addition, the doppler

spread is dependent on the mobile speed, the delay spread is dependent on

the distance of scatterers relative to the base or mobile, and the angle spread

is dependent on the relative height of the scatterers with the antennas. There-

fore these channel statistics are dependent on the characteristics of the users

and on the base station antenna array architecture. The shadowing of a chan-

nel captures longer-term characteristics of the channel that depend on the

location of large-scale terrain features like buildings and mountains as well

as on the large-scale density and location of scatterers.

User characteristics

The network provides service to a population of users that is characterized

by a spatial distribution, a velocity distribution and service requirements.

A realization of the spatial distribution gives the location for a user, and a

realization of the velocity distribution gives its speed. Only a subset of the

users are actively transmitting or receiving data while the rest of them are

in an idle state. The service requirements for each user depend on the appli-

cation, such as voice, data downloading, streaming media, or gaming. These

characteristics can be parameterized by their average data rate and latency

requirements.

Fundamental constraints

Fundamental constraints for cellular network design are based on regulations

for using the licensed spectrum. An operator purchases bandwidth on certain

carrier frequencies and is allowed to transmit with a limited effective radiated

power. The spectrum is typically sold in paired bands for frequency-division

duplexing (FDD) or unpaired bands for time-division duplexing (TDD).

Cost constraints

Monetary costs associated with network design include capital expenses for

developing and deploying network infrastructure (base stations and back-

haul) and operational expenses that include power, rent on real estate, main-

tenance, and backhaul [136]. The computational complexity of signal pro-

cessing and network management algorithms has an impact on the cost of

hardware and software implementation.

5.1 Cellular network design components 163

MAC and PHY layer design

The physical layer defines the specifications for the air interface, including

the frame structure, options for modulation and coding, the multiple-access

technique, and the MIMO transceiver techniques. The MAC layer determines

how the PHY-layer resources are accessed. In packet-switched networks, the

base station allocates the resources using a scheduler (Section 5.5.4) for both

uplink and downlink transmission.

Base station design

The location of the base stations and the design of each base are critical for

providing adequate coverage and capacity in the network. The base design

includes the number of sectors per site and the antenna array architecture

for each sector. The array consists of multiple antenna elements, and each

element can be characterized by its beamwidth (horizontal and vertical) and

the polarization (vertical, slant, cross-polarized). The physical dimensions of

the antenna element are based on these characteristics and related to the

carrier frequency as described in Section 5.5.1. The array is characterized by

the positions and orientation of the antenna elements. For example, elements

could be arranged in a linear configuration with close (half-wavelength) spac-

ing or diversity (many wavelengths) spacing, as described in Section 5.5.1.

The downtilt angle, the vertical position, and the azimuthal orientation of

the array elements could be adjusted to affect the coverage.

The location and height of base station antennas are often constrained by

zoning restrictions and must conform to local aesthetic values. These restric-

tions become more challenging as the number of antennas increases and the

size of the array grows. Larger arrays also require more robust infrastructure

to compensate for the additional weight and windloading effects.

The bases in a cellular network are connected to the core network through

backhaul links. The core network provides access to the Internet or conven-

tional phone network. The backhaul links are often wired connections run-

ning over copper wire or fiber optic cable, but they could also be implemented

wirelessly on spectral resources orthogonal to those used for serving mobile

terminals.

Terminal design

The mobile terminal consists of the signal processing capabilities and the

antenna configuration. Most mobile handsets support one or two antennas.

Larger mobile devices such as tablets or laptops have more real estate and


computational power to support more antennas and more sophisticated al-

gorithms.

Starting with a blank slate, an engineer would design the entire system—PHY

and MAC layers, base stations, and terminals—from scratch. For a mature

network where the base station infrastructure is already deployed, however,

an engineer would have limited means of optimizing performance. Air in-

terface parameters could be altered through software, and the downtilt and

orientation of base station antennas could be adjusted to optimize coverage.

The context of the MIMO design strategies discussed in this and the follow-

ing chapter lies between these two extremes. We are interested in evaluating

MIMO techniques within the framework of an existing packet-based cellular

network standard. Therefore, with regard to Figure 5.1, we assume the MAC

and PHY layers are fully defined for SISO links, and we focus on the impact

of the MIMO techniques and the antenna array architecture at the base and

mobile. We assume that a generic OFDMA air interface partitions the band-

width into parallel subbands so we can study the performance of MIMO in

a narrowband channel.

5.2 System model

In practice, the location of base stations is irregular, and dependent on factors

such as the terrain, zoning restrictions and user distribution. However, for

the purposes of idealized performance evaluation, it is convenient to assume

a regular placement of bases as shown in Figure 5.2, where the geographic

area is partitioned into a hexagonal grid of cells and where a base is at the

center of each cell site. As a result of different transmission powers and signal

fading, the base closest to a user is not necessarily the one assigned to it.

Therefore a user lying within a given cell is not necessarily assigned to that

base, and the cell boundaries simply highlight the placement of the bases.

Each site is partitioned into S sectors, and a base station is associated with

each sector. Each base is equipped with M antennas and serves K users, each

with N antennas. The uplink received signal at base b (b = 1, . . . , B) is

xb =

KB∑k=1

Hk,bsk + nb (5.1)

5.2 System model 165

=∑j∈Ub

Hj,bsj +∑j /∈Ub

Hj,bsj

︸︷︷︸interference

+ nb, (5.2)

where

• Hk,b ∈ CM×N is the block-fading MIMO channel between the kth user

(k = 1, . . . ,KB) and the bth base

• sk ∈ CN is the transmitted signal from user k

• nb ∈ CM represents receiver thermal noise with distribution CN (0M , σ2IM )

Equation (5.2) separates the received signal in terms of desired signals from

users assigned to base b (denoted by the set Ub) and interference signals from

all other users. The transmit covariance for user k is Qk := E(sksHk ), with

a power constraint tr(Qk) ≤ Pk. (As we will describe in Section 5.2.2, this

is actually a scaled power constraint that does not correspond to the actual

maximum allowable radiated power.)

The downlink signal received by the kth user (k = 1, . . . ,KB) is

xk =

B∑b=1

HHk,bsb + nk (5.3)

= HHk,b∗sb∗ +

∑b =b∗

HHk,bsb︸︷︷︸

interference

+ nk, (5.4)

where

• HHk,b ∈ C

N×M is the block-fading MIMO channel between the kth user

and the bth base (b = 1, . . . , B)

• sb ∈ CM is the transmitted signal from base b

• nk ∈ CN is the thermal noise vector with distribution CN (0N , σ2IN )

Under the assumption that user k is assigned to base b∗, equation (5.4) sep-

arates the received signal in terms of the desired and interfering signals. The

transmit covariance for base b is Qb := E(sbsHb ), with a (scaled) power con-

straint tr(Qb) ≤ P .

For both the uplink and downlink, we assume the components of the chan-

nel Hk,b are i.i.d. Rayleigh with distribution h(m,n)k,b ∼ CN (0, α2

k,b). The vari-

ance α2k,b can be interpreted as the average channel gain between user k and

base b relative to a user located at a reference distance dref from the base.

This value depends on the distance-based pathloss, shadow fading realization,

and direction between user k and base b:


……

…

…

…

…

S = 3 sectors per siteM = 4 antennas per sector

Broadside direction of the M antennas

Fig. 5.2 Network of hexagonal cells. Each cell site is partitioned radially into S sectors,

and each sector uses M antennas to serve K users.

α2k,b =

(dk,bdref

)−γ

Zk,bGk,b, (5.5)

where

• dk,b is the distance between user k and base b

• dref is the reference distance with respect to which the pathloss is mea-

sured, described in Section 5.2.2

• γ is the pathloss exponent

• Zk,b is the shadowing realization between user k and base b caused by

large-scale obstructions such as terrain and buildings

• Gk,b is the direction-based antenna response of the base antenna, described

in the following section

The pathloss exponent value is γ = 2 for free space, but it is in the range of

3 to 4 for typical cellular channels. In the spatial channel model often used in

3GPP cellular standards, γ = 3.5 for suburban and urban macrocells with a

carrier frequency of 1.9 GHz. The γ value was derived from a modified Hata

urban propagation model which is applicable over a frequency range from 1.5

to 2.0 GHz [36]. The next-generation mobile networking (NGMN) simulation

methodology [135] assumes that γ = 3.76 for a carrier frequency of 2.0 GHz.

In macrocellular outdoor environments, the shadowing realization is typi-

cally modeled with a log-normal distribution with standard deviation 8 dB:


Z = 10(x/10), x ∼ η(0, 82). The shadowing realization could be correlated

between users (or bases) located near each other if they are similarly affected

by common obstructions.

5.2.1 Sectorization

To implement sectorization, base antennas have a directional response that

varies as a function of the angular (azimuthal) direction. The response Gk,b

for a directional antenna element is a function of the angular difference be-

tween the pointing (broadside) direction of the antenna and the direction

of the user with respect to this element. As shown in Figure 5.3, ωb is the

broadside direction of sector antenna b, and θk,b is the direction of user k with

respect to the location of sector b. The angles are measured with respect to a

common reference (the positive x-axis). We will assume that multiple anten-

nas belonging to a given sector have the same pointing direction and antenna

response, but this assumption is not necessary in general.

When there is no sectorization, the bases employ omni-directional base

antennas whose response is unity independent of the user location: Gk,b = 1.

Under sectorization, we assume that each cell site is divided radially into S

sectors. The broadside directions of the sector antennas is shown in Figure

5.4. For S = 3, the broadside directions of the three sectors are 90, 210, and

330 degrees, giving the so-called “cloverleaf” pattern of sector responses. For

S = 6 and 12 sectors, the broadside direction of the sth sector (s = 1, . . . , S)

is 360(s−1)S degrees.

Under ideal sectorization, the antenna gain is unity within the sector and

zero outside:

Gk,b(θk,b − ωb) =

{1 if |θk,b − ωb| ≤ π

S

0 if |θk,b − ωb| > πS .

(5.6)

In normalizing the antenna gain to be unity within the sector, we implicitly

assume that the total transmit power per site is fixed and that the reduction

in power per sector is offset by the gain achieved with the narrower beam

pattern. In other words, with S sectors per site, each sector transmits with a

fraction 1/S and achieves a directional gain of factor S. These factors cancel

one another so the gain is unity, independent of S.

In practice, there is intersector interference as a result of non-ideal antenna

patterns. The response used in simulations is often a parabolic model that


closely matches the response of commercial sector antennas:

Gk,b(θk,b − ωb)(dB) = max

{−12

[(θk,b − ωb)

Θ3 dB

]2, As

}, (5.7)

where Θ3 dB is the 3 dB beamwidth of the response and As < 0 is the sidelobe

level of the response, measured in dB. This parabolic response is sometimes

known as the Spatial Channel Model (SCM) sector response [36]. Note that

the response Gk,b(θk,b − ωb) = −3 dB when the user angle is half the 3 dB

beamwidth: θk,b − ωb = Θ3 dB/2. As with the ideal sector response, the re-

sponse in (5.7) is normalized under the assumption of fixed transmit power

per site. The sector responses for ideal, parabolic, and commercially available

antennas are shown in Figure 5.6 for the case of S = 3 sectors per site. The

parabolic response, with Θ3 dB = 70π/180 and As = −20 dB, is a good ap-

proximation of the commercial response over the main lobe and overestimates

the sidelobe level. Therefore the simulated performance will overestimate the

interference and result in a lower bound on capacity performance.

The 3 dB-beamwidth and sidelobe level parameters are dependent on the

number of sectors per cell S. To ensure reasonable intersector interference,

the beamwidth and sidelobe levels should decrease as the number of sectors

per cell increases. The parameters we use in our simulations are summarized

in Figure 5.5. We will refer to a site with more than S = 3 sectors as having

higher-order sectorization. In going from S = 3 to 6 sectors, the beamwidth is

reduced by a factor of two, and the antenna gain increases by 3 dB. However,

because we fix the total transmit power per site, the power for each of the

S = 6 sectors is halved. The antenna gain and power reduction cancel each

other, so the response in the broadside direction is unity, and the sidelobe

levels drop by 3 dB. The same scaling occurs in going from S = 6 to S = 12

sectors. The responses for S = 3, 6, 12 sectors per site are shown in Figure

5.7 for the parameters in Figure 5.5. We emphasize that these parameters

are chosen to model well-designed sectorized antennas. Antenna responses of

actual sectorized antennas may differ from this parabolic approximation.

The parabolic response is an approximation of an actual antenna’s re-

sponse measured in an anechoic chamber with zero angle spread. Because

of the non-zero angle spread observed in practice, signals transmitted from

the base will experience angular dispersion, resulting in a wider antenna re-

sponse. The resulting antenna response can be obtained by convolving (in

the angular domain) the channel’s power azimuth spectrum (PAS) with the


zero-angle-spread response in (5.7) [137]. The power azimuth spectrum of

a typical urban macrocellular channel can be modeled as a Cauchy-Lorenz

distribution with an RMS angle spread of φ = 8π/180 radians [138]. The rel-

atively narrow angle spread is due to the fact that the base station antennas

are higher than the surrounding scatterers. Figure 5.8 shows the resulting

antenna responses for S = 3, 6, 12 sectors. Because the PAS is relatively nar-

row compared to the baseline S = 3 response in Figure 5.7, the resulting

convolved response is similar to the baseline response. However, the PAS is

wide compared to the S = 12 baseline response, and the resulting convolved

response is significantly wider than the baseline response.

Broadside direction

…

User

antennas

…

Fig. 5.3 The direction of user k with respect to the M antennas of sector b is given by

θk,b. The broadside direction of the M antennas is ωb. These directions are measured with

respect to the positive x-axis.

5.2.2 Reference SNR

The reference SNR is a single parameter that captures the effects of link

budget parameters such as transmit power, antenna gain, cable loss, receiver

noise, additive noise power spectral density, and bandwidth. Encapsulating

these parameters is very useful because it allows us to present simulation

results using a single parameter instead of a list of parameters.


hh

Fig. 5.4 Broadside direction of sectorized antennas for S = 3, 6, 12 sectors per cell site.

The downlink reference SNR is the average SNR of a signal transmitted

from a reference base and measured by a user at a reference location, as shown

in Figure 5.9. The following assumptions are made in defining the reference

SNR.

• The base transmits with full power P

• The user is is located at the reference location at the cell border with

distance d = dref from the reference base

• The user is in the boresight direction of the base so that the directional

antenna response is G = 1

• The shadow fading realization is Z = 1

Under these conditions, the average channel gain (5.5) is α2 = 1. The received

signal is x = hs+ n, and the average SNR is

E[|h|2]E [|s|2]E [|n|2] =

α2P

σ2=

P

σ2. (5.8)

Therefore the downlink reference SNR is simply Pσ2 . Because the SNR is

measured at the cell edge, the reference SNR is sometimes known as the cell-

edge SNR. While we have considered the SISO channel here, the reference

SNR does not depend on the number of transmit antennas if the channel is

i.i.d. and if the total transmit power is constrained to be P .

For the uplink, we assume that the total power constraint of the K users

assigned to a base is P and that the power constraint for each user is the

same: Pk = P/K, for k = 1, . . . ,K. We define the uplink reference SNR as


0

(dB

)

(degrees)

-3

0

Sectors per site( )

3612

(degrees)7035

17.5

(dB)-20-23-26

Fig. 5.5 Parabolic sector response (5.7). The response is parameterized by the 3 dB

beamwidth (Θ3dB) and sidelobe level (As).

-200 -150 -100 -50 0 50 100 150 200-30

-25

-20

-15

-10

-5

0

θ - ω (degrees)

G( θ

- ω) (

dB)

ParabolicIdealCommercial

Fig. 5.6 Sector responses for S = 3 sectors per site. The parabolic response is from (5.7),

and the ideal response is given by (5.6). The parabolic response matches the response of the

commercially available antenna over the main lobe and overestimates the sidelobe level.


-200 -150 -100 -50 0 50 100 150 200-30

-25

-20

-15

-10

-5

0

θ - ω(degrees)

G( θ

- ω) (

dB)

S = 3S = 6S = 12

Fig. 5.7 Parabolic responses for S = 3, 6, 12 sectors per site, given by (5.7) and parameters

in Figure 5.5.

-200 -150 -100 -50 0 50 100 150 200-30

-25

-20

-15

-10

-5

0

θ - ω (degrees)

G( θ

- ω) (

dB)

S = 3S = 6S = 12

Power azimuth

spectrum

Fig. 5.8 Sector responses for S = 3, 6, 12 sectors per site obtained by convolving the

power azimuth spectrum (for 8 degree angle spread) with the parabolic response in Figure

5.7.


the average total SNR received by the reference base when the K users at

the reference location transmit with full power: Pσ2 .

Recall that for the SU- and MU-MIMO channels, the SNR is similarly

defined as Pσ2 . If the isolated channels are normalized to have unity gain

E[|h|2i,j] = 1, then P is both the transmit power constraint and the average

received power. For the downlink cellular channel, we see now that P is the

power constraint scaled with respect to the reference location. The actual

radiated power constraint can be determined from P by accounting for the

channel loss from the reference base to reference user location.

Reference user Reference base

Fig. 5.9 The reference SNR P/σ2 is measured by a user located in the boresight direction

of the serving base at distance dref , which is half the intersite distance.

The reference SNR depends on the link budget parameters, and is usually

different for the uplink and downlink. Examples of reference SNR values and

their associated parameters are given in Figure 5.10. The parameters for the

two downlink examples have been chosen to give respective guidelines for

high and low reference SNRs. Downlink A is an example of a base station

transmitter with relatively high power (40 watts) and its parameters corre-

spond to the NGMN model. The reference received signal power and noise

power are respectively

P = 46 dBm− 9 dB + 14 dB− 128 dB + 37.6 log10

(1000

250

)dB = −54.3 dBm

σ2 = −174 dBm+ 10 log10(107) dB = −104 dBm,

and the reference SNR is P/σ2 = 49.6 dB. The NGMN downlink simulation

parameters are similar to those given by Downlink A, except that there is a

20 dB in-building loss, resulting in a cell-edge SNR of 29.6 dB. Downlink B is

a base station with lower transmit power and larger inter-site distance. With


an intersite distance of 2000 m, each cell has an area of about 3 km2, and

covering a large city with an area of 1000 km2 would require several hundred

bases. Downlink B also includes a 20 dB in-building penetration loss, resulting

in a reference SNR of 1.4 dB. Because of the significant attenuation of signals

passing through walls, indoor coverage is achieved more effectively by using

indoor bases.

Parameters for Uplink A and B give respective upper and lower guide-

lines for the uplink reference SNRs. Except for transmit powers and receiver

noise figures, the parameters are the same for both uplink and downlink.

Typically, transmit powers are higher for the downlink because base stations

have larger power amplifiers, and receiver noise is also higher in the downlink

due to noisier electronics at the handset. The NGMN uplink parameters are

similar to Uplink B, except that the reference distance is 250 m, resulting in

a reference SNR of 11.6 dB.

The pathloss intercept has a logarithmic dependency on the carrier fre-

quency f (measured in units of mega-Hertz) given approximately byB log10 f ,

where the parameter B depends on parameters such as the antenna height

and terrain type. Its value is approximately 30 for typical macrocellular de-

ployments [36] so a doubling of the carrier frequency from 1000 MHz to

2000 MHz would result in an additional pathloss of about 10 dB. Therefore

to maintain the same coverage for higher carrier frequencies, higher transmit

power or higher antennas are required.

5.2.3 Cell wraparound

For a given downlink cellular network, users at the edge of the network will

experience less interference than those at the center. In order to make the

interference statistics the same regardless of user location, we can approxi-

mate an infinite cellular network by projecting a finite network on a torus.

Cells at one edge of the network are “wrapped around” the torus to be adja-

cent to cells at the opposite edge. A wrapping strategy based on a hexagonal

network of hexagonal cells has desirable characteristics because it exhibits

translational and rotational invariance [139].

Figure 5.11 shows a hexagonal wrapping using a network of B = 7 cells

labeled 1a, 2a,..., 7a. This network is replicated six times around the perime-

ter. The neighbor to the northwest of cell 6a is cell 4e, which is a replica


Downlink A46 dBm(40 W)

-174 dBm/Hz

10 MHz9 dB14 dB0 dB

128 dB3.76

500 m250 m

49.6 dB

Transmit power

Noise power spectral density ( )

Bandwidth ( ) Receiver noise figure

Net antenna gainIn-building penetration loss Pathloss intercept, 1000m

Pathloss exponent ( )Intersite distance ( )

Reference distance ( )

Reference SNR

Downlink B38.6 dBm(7.2 W)

-174 dBm/Hz

10 MHz9 dB14 dB20 dB

128 dB3.76

2000 m1000 m

0.0 dB

Uplink A30 dBm(1 W)

-174 dBm/Hz

10 MHz5 dB14 dB0 dB

128dB3.76

500 m250 m

37.6 dB

Uplink B24 dBm

(250 mW)

-174 dBm/Hz

10MHz5 dB14 dB0 dB

128 dB3.76

2000 m1000 m

-11.0 dB

Fig. 5.10 Examples of link parameters and corresponding reference SNRs. Downlink A

and Uplink A are relatively high reference SNRs because they use high-powered transmit-

ters with small intersite distances. Downlink B and Uplink B are relatively low reference

SNRs because they user low-powered transmitters with large intersite distances and a

significant in-building penetration loss.

of cell 4a. In the simulations, users are placed with a uniform distribution

over the original network of B cells. Using cell wraparound, the channel gain

α2k,b between a user k and a given base b = 1, . . . , B is the maximum of the

channel gains between the user and the seven wraparound replicas of base b,

where the shadowing realization for all replicas is the same. For example, the

channel gain between user k and cell 1 is

α2k,1 = max

[α2k,1a, α

2k,1b, . . . , α

2k,1g

](5.9)

= max

[(dk,1adref

)−γ

Zk,1Gk,1a, . . . ,

(dk,1gdref

)−γ

Zk,1Gk,1g

]. (5.10)

Repeating this procedure for all B cells, we see that the channel variances

for a given user placement are, in effect, measured with respect to the B

cells with the minimum radio distance. For the case of omni-directional base

antennas, Gk,1a = · · · = Gk,1g = 1, and

α2k,1 =

[min (dk,1a, dk,1b, . . . , dk,1g)

dref

]−γ

Zk,1. (5.11)


In this case, the channel gains are measured with respect to the B cells with

the minimum Euclidean distance. For example in Figure 5.11, the B channel

gains for a user placed in cell 6a are with respect to the cells highlighted by

the black border. For cell wraparound to be effective, the size of the network

should be sufficiently large that the channel gain of the dominant replica in

(5.9) is several orders of magnitude larger than the others. For a pathloss

coefficient of γ ≈ 4, a network of B = 19 cells is sufficiently large. (We used

a smaller network with B = 7 cells just for illustrative purposes.) While we

have described the wrapping technique for the downlink, it can be applied to

the uplink in a similar fashion.

1b

6b

7b

2b

5b

3b

4b

1c

6c

7c

2c

5c

3c

4c

1f

6f

7f

2f

5f

3f

4f

1e

6e

7e

2e

5e

3e

4e

1g

6g

7g

2g

5g

3g

4g

1a

6a

7a

2a

5a

3a

4a

1d

6d

7d

2d

5d

3d

4d

Fig. 5.11 Network of B = 7 cells with cell wraparound. The original cells (1a, 2a,...,7a)

are shaded gray. The B channel gains for each user are measured with respect to the

network of B cells with the smallest radio distance. With omni-directional base antennas,

the smallest radio distance becomes the smallest Euclidean distance. For example, a user

placed in cell 6a measures its channel gain with respect to the B cells highlighted by the

black border.

5.3 Modes of base station operation

In conventional cellular networks, communication within each cell is indepen-

dent of the other cells. Interference caused to other cells is treated as noise,

5.3 Modes of base station operation 177

and if the interference is stronger than the thermal noise, the performance

becomes interference limited. As mentioned in Chapter 1, interference can

be mitigated in the time and frequency domains. In this section, we describe

how interference can be mitigated in the spatial domain using the MIMO

techniques described in previous chapters. The efficiency of the mitigation

depends on the mode of base station operation. Base stations can operate

independently or in a coordinated fashion. These options are highlighted in

Figure 5.12 and described below.

Downlink Uplink

Coo

rdin

ated

bas

esIn

depe

nden

t bas

es

Inte

rfere

nce-

obliv

ious

Inte

rfere

nce-

awar

e

Network MIMO Network MIMO

Coordinated precodingInterference alignment

Fig. 5.12 Modes of operation for cellular networks. Solid lines indicate signals to assigned

users (downlink) or from assigned users (uplink). Dashed lines indicate signals that cause

interference. An “x” over a dashed line indicates significantly mitigated interference. The

double arrows indicate coordination between bases.


5.3.1 Independent bases

Under independent base station operation, no information is shared be-

tween the bases. Interference in the spatial domain can be mitigated in an

interference-oblivious or interference-aware manner.

Interference-oblivious mitigation

Interference-oblivious mitigation refers to spatial processing performed at a

transmitter or receiver which does not account for the channel state informa-

tion (CSI) of, respectively, unintended receivers or interfering transmitters.

In an uplink SIMO network where each user has a single antenna and where

the base has M antennas, interference-oblivious mitigation is implemented

with a matched filter (maximal ratio combining) receiver. Let us consider an

uplink SIMO channel (5.2) with the received signal at the assigned base given

by

x = hksk +∑j =k

hjsj + n, (5.12)

where k is the index of the desired user and where the signals from all other

users j �= k cause interference. The matched filter receiver output hHk x has

an SINR given by (2.58), and the achievable rate is

R = log2

(1 +

‖hk‖4 P‖hk‖2 σ2 +

∑j =k

∣∣hHk hj

∣∣2 Pj

). (5.13)

If there is no interference (Pj = 0 for all j �= k), then the signal is corrupted by

spatially white noise, and the matched filter is optimal (Section 2.3.1). Due to

the presence of intercell interference, the matched filter is no longer optimal.

However, even though this receiver has no knowledge of the interferers’ CSI

(hj , j �= k), interference can be mitigated if a significant fraction of the power

lies in the null space of the desired channel hk. For example, interference

is totally mitigated if interference channels are orthogonal to the desired

channel: hHk hj = 0, for all j �= k.

For a given channel hk, the statistics of the rate (5.13) depend on the

relationship between hk and the distribution of the interferers’ channels

(hj , j �= k). For the case of i.i.d. Rayleigh channels, the average rate can

be written as:


Ehj(R) ≥ log2

(1 +

‖hk‖4 P‖hk‖2 σ2 +

∑j =k Ehj

∣∣hHk hj

∣∣2 Pj

)(5.14)

= log2

(1 +

‖hk‖4 P‖hk‖2 σ2 +

∑j =k ‖hk‖2 α2

jPj

)(5.15)

= log2

(1 + ‖hk‖2 Pk

σ2 +∑

j =k α2jPj

), (5.16)

where (5.14) follows from Jensen’s inequality applied to the convex function

f(a) = log2(1 + (1 + aHRa)−1) and where (5.15) follows from the assump-

tion of i.i.d. Rayleigh channels. The expression in (5.16) corresponds to the

capacity of a SIMO channel hk where the interference is spatially white with

covariance IN (σ2 +∑

j =k α2jPj). Therefore by using a matched filter, the av-

erage rate achievable in the presence of colored interference is at least as high

as the rate achievable in the presence of spatially white interference with the

same power.

For the downlink, interference-oblivious mitigation for MISO channels can

be implemented using maximal ratio transmission (Section 1.1.2.3) if the base

station has knowledge of the desired user’s channel. Similar to the uplink, in-

terference can be implicitly mitigated if the channels to the unintended users

are not aligned with the desired user’s channel.

Interference-aware mitigation

Under interference-aware mitigation at the receiver, the spatial processing

uses explicit channel knowledge of the interfering transmitters. For example,

for the uplink SIMO channel (5.12), interference-aware mitigation for the de-

sired user k can be implemented with an MMSE receiver (Section 2.3.1) using

knowledge of hk and the interfering users’ channels hj , j �= k. The receiver

does not make any distinction between interfering users assigned to the base

serving user k or assigned to other bases. CSI estimates could be obtained by

demodulating pilot signals sent by each user. Since each user would transmit

a pilot signal for coherent demodulation by its assigned base, interference-

aware mitigation at the receiver can be achieved with no additional signaling

overhead.

Because the MMSE receiver uses explicit knowledge of the interferers’

channels, the MMSE receiver is more efficient than the matched filter for

mitigating interference. Using the expression for the SINR at the output of

the MMSE receiver (2.61), the achievable rate is


R = log2

⎡⎢⎣1 + Pk

σ2hk

⎛⎝IM +

∑j =k

Pj

σ2hjh

Hj

⎞⎠−1

hk

⎤⎥⎦ . (5.17)

Because the MMSE receiver maximizes the SINR among all linear receivers,

its output SINR will be greater than that of the matched filter, and its rate

(5.17) will be greater than that of the matched filter (5.13) for any set of

channel realizations. It follows that for a fixed channel hk, the MMSE rate

(5.17) averaged over the interfering channels will be greater than the lower

bound for the average MF rate (5.16).

Interference-aware mitigation can also be implemented for MIMO channels

using a pre-whitening filter. (We recall that the MMSE receiver for a mul-

tiuser SIMO channel is equivalent to pre-whitening the colored interference

followed by matched filtering.) We consider an uplink MIMO channel (5.2)

where single-user MIMO is implemented. The received signal at the assigned

base for user k is

x = Hksk +∑j =k

Hjsj + n, (5.18)

where the channels Hj , j �= k are for the interfering users assigned to other

bases. The transmit covariances for these users Qj = E(sjsHj ), j �= k are

set according to the channels of their respective serving bases. Assuming the

interference covariance

QI := σ2IM +∑j =k

HjQjHH (5.19)

is known by the base serving user k, the output of the pre-whitening filter is

Q−1/2I x. If the transmit covariance for the desired user is Qk, the achievable

rate is

R = log2 det[IM +Q−1

I HkQkHHk

]. (5.20)

In a similar manner, pre-whitening can be applied at the mobile receiver for

downlink single-user MIMO transmission, and more generally, pre-whitening

can also be applied for multiuser MIMO channels. For example, given the

uplink received signal at base b (5.2), the pre-whitening is performed with

respect to the interference covariance

σ2IM +∑j /∈Ub

Hj,bQjHHj,b. (5.21)


On the downlink, given the received signal by user k (5.4), the pre-whitening

is performed with respect to the interference covariance

σ2IN +∑b=b∗

HHk,bQbHk,b. (5.22)

Interference-aware mitigation can be implemented at the transmitter if the

channels of the unintended receivers are known. However, acquiring reliable

channel estimates may be difficult in practice, especially for downlink trans-

mission in FDD systems (Section 5.5.4). Linear precoding techniques such as

those described in Section 4.2.1 could be used. In general, more sophisticated

non-linear techniques could be used for interference-aware mitigation at the

transmitter or receiver.

5.3.2 Coordinated bases

Compared to using independent bases, interference can be mitigated to a

greater extent by coordinating the transmission and reception among groups

of spatially distributed base stations. Coordination requires backhaul between

bases to have higher bandwidth in order to support the sharing of informa-

tion among them. Depending on the coordination technique, this information

could consist of CSI, baseband signals, and/or user data bits. The bottom

half of Figure 5.12 summarizes the coordination techniques which are de-

scribed below.

Coordinated precoding

When bases operate independently, the precoding at a given base is a func-

tion of the local CSIT, which refers to the channels between the base and

the surrounding users. With coordinated precoding, CSIT is shared between

bases so that the precoding at a given base could be a function of CSIT as

seen by multiple bases. (The implementation of CSIT sharing is described

below in Section 5.5.5.) Most generally, the precoding at each base could be

a function of the global CSIT, meaning the CSIT between all bases and all

users. The transmitted signal at each base requires global CSIT but only

local data, meaning each base transmits data only to users assigned to it.

Global CSIT allows the precoding for all the bases to be computed jointly to

optimize the performance criterion.


A special case of coordinated precoding for the case of directional beams

is coordinated beam scheduling. As shown in Figure 5.13, global CSIT would

allow each base to realize the potentially hazardous situation on the left

where co-located users are assigned to different bases but served on the same

time-frequency resources. To avoid this situation which would result in high

interference for all users, the bases would schedule the co-located users on

different resources.

AA A

B B

High interference

A B

B A

Low interference

Fig. 5.13 On the left, users in each pair of co-located users are served by different bases

but on the same time-frequency resource, resulting in high interference. Using coordinated

beam scheduling, the bases would perform joint scheduling so that co-located users are

served on different resources.

Interference alignment

Under interference alignment, precoding is implemented at each base so that

the interference at each terminal is aligned to lie in a common subspace

with minimum dimension, and the remaining dimensions can be used for

interference-free communication with the assigned base. Like coordinated

precoding, interference alignment requires global CSIT for precoding at each

base. However unlike coordinated precoding, which can be implemented on

a frame-by-frame basis, precoding for interference alignment occurs over an

infinite number of independently faded time or frequency realizations [140].

As an example, we consider an interference channel with three single-

antenna bases, each communicating with a single-antenna user as shown in

Figure 5.14. Each user attempts to decode the signal from its assigned base in

the presence of interference from the other two bases. Communication occurs

over F time frames. We let H[f ] represent where the 3 × 3 channel during

frame f = 1, . . . , F , and the channel is assumed to be spatially and tempo-

rally rich so the distribution is i.i.d. Rayleigh with respect to time and space.

Interference alignment uses precoding (across time) at each transmitter such

that as F goes to infinity, an average of 3/2 streams per frame are received


interference-free by the users. In general, for a system withK transmit-receive

pairs (where each node has a single antenna), interference alignment asymp-

totically achieves K/2 interference-free streams per frame [140]. This K/2

factor is the multiplexing gain of the sum-capacity at high SNR. For compar-

ison, under TDMA transmission each of K transmit/receive pairs would be

active for a fraction 1/K of the time, achieving a multiplexing gain of only 1.

If bases transmit simultaneously and independently, each user receives inter-

ference from the other bases, and the multiplexing gain is zero. If the channels

among the K pairs were mutually orthogonal, a multiplexing gain of K could

be achieved. Hence interference alignment enables noise-limited performance

and achieves half the degrees of freedom of an interference-free channel.

In practice, interference alignment is difficult to implement because it re-

quires very large time (or frequency) expansion and very rich channels. An

alternative technique for static channels requires infinite resolution of channel

realizations [141]. Techniques for achieving the “K/2” degrees of freedom in

practical scenarios is an area of active research.

s1[1]…s1[F]

s2[1]…s2[F]

s3[1]…s3[F]

H[1]…H[F]

Fig. 5.14 Interference alignment for an interference channel with three transmit/receive

pairs. Interference at each receiver is aligned to lie in a minimum-dimensional “garbage”

subspace so the remaining dimensions can be used for interference-free communication

with the intended transmitter. In this example, one stream per pair can be communicated

interference-free.


Network MIMO

Under network MIMO, base stations operate in a fully coordinated manner

as if they belonged to a single virtual base with distributed antennas. For

example on the downlink, a coordination cluster of B base stations, each with

M antennas, operates over a broadcast channel with BM transmit antennas

[142]. The data for each user is in general transmitted from all bases in the

cluster, and the precoding at each base depends on the global CSIT. Therefore

downlink network MIMO requires both global CSIT and global data of all

users at each base, resulting in higher backhaul bandwidth requirements than

coordinated precoding and interference alignment. Downlink network MIMO

also requires tight synchronization and phase coherence among coordinated

bases so that signals transmitted from different bases arrive at each user with

the proper phase.

Network MIMO can be implemented on the uplink in a similar fashion

by using multiuser detection to jointly detect users across multiple bases

[143]. A cluster of B coordinated bases, each with M antennas, implementing

network MIMO could be viewed as a single virtual base with BM distributed

receive antennas. One could therefore implement uplink network MIMO by

collecting the baseband signals from a cluster of bases and performing joint

detection (for example, MMSE-SIC) at a central location. For a system with

multiple-antenna users, scale-optimal multiplexing gain can be achieved if

the number of users exceeds BM by transmitting a single stream per user

without precoding or by using only local CSIT at each mobile (Section 4.1.1).

Therefore significant performance gains using uplink network MIMO can be

achieved without global CSIT knowledge, making it much easier to implement

than downlink network MIMO. As a lower-complexity alternative to joint

detection, one could selectively cancel interference by exchanging decoded

data of dominant interferers among cooperating cells and reconstructing the

interference signals. This technique achieves similar performance gains to

joint detection network MIMO for low-geometry users but with significantly

reduced backhaul overhead [144].

Figure 5.15 shows different modes of coordination for downlink network

MIMO. (These modes apply to the uplink as well.) With non-overlapping

clusters, the network is partitioned into fixed coordination clusters. Each user

is assigned to a single cluster and is served by the bases belonging to that

cluster. In the left subfigure of Figure 5.15, the network is partitioned into two

clusters of B = 2 bases each. Users will experience interference from adjacent

clusters, and those at the edge of the cluster will experience more interference


Full-network coordinationOverlapping clustersNon-overlapping clusters

Fig. 5.15 Coordination modes for network MIMO. Interference occurs between clusters.

Under full-network coordination, all bases in the network are coordinated and, under ideal

conditions, performance is limited by thermal noise.

than those near the center. To reduce the intercluster interference, one could

form user-specific clusters of B bases so that each user is in the center of

its assigned cluster. In general, different clusters consist of different sets of

bases, and a given base may belong to multiple clusters. Different clusters that

consist of at least one common base are said to overlap, as illustrated in the

center subfigure of Figure 5.15 where two clusters of B = 3 bases overlap and

share two common bases. For a given cluster size B, the user-specific clusters

are more general than the fixed clusters; hence the geometry statistics with

user-specific, overlapping clusters could be better than those with fixed, non-

overlapping clusters. Network MIMO with overlapping clusters is described

in Section 5.4.4 for minimum sum-power transmission, and it is addressed

in [145] for full-power transmission. A survey of techniques for network MIMO

with clustering is given in [146].

With either non-overlapping or overlapping clusters, the system perfor-

mance is limited by intercluster interference. However, if all the bases in the

entire network are coordinated so that all users are served by a single coor-

dination cluster, the network performance would be noise limited. Network-

optimal performance could be achieved if capacity-achieving strategies (DPC

on the downlink and MMSE-SIC on the uplink) are ideally implemented

across the entire network [142]. A survey of analytical performance results

for full network coordination in the context of the Wyner model can be found

in [146]. Most of these results assume ideal coordination and perfect knowl-

edge of global CSI. In practice, accurate CSIR is difficult to obtain, especially


for larger networks where separation between transmitters and receivers is

large.

For low-mobility users, the CSI quality is sufficiently good to support full

network coordination for a small indoor network [147]. However, for larger

networks, acquiring reliable global CSI and sharing the information across

the network becomes a significant challenge.

Figure 5.16 summarizes the features of downlink MIMO options in the context

of a network with B bases. Each base b = 1, . . . , B is equipped with M

antennas and is assigned to an M -antenna user with index b. The data signal

for user b is denoted as ub. The channel between base b and user k is HHk,b.

For interference-oblivious precoding with closed-loop SU-MIMO, base b

requires knowledge of only user b and its channelHHb,b. If the spectral resources

are partitioned into B mutually orthogonal blocks and if each base transmits

on one of these blocks, a multiplexing gain of M/B per base can be achieved.

Under interference-aware precoding, base b requires local CSIT with respect

to all users HHk,b, k = 1, . . . , B. Both coordinated precoding and interference

alignment require global CSIT for each base b: HHk,j , j, k = 1, . . . , B. The

multiplexing gain per base with interference alignment is M/2 [140] in the

limit of an infinite time horizon. This gain is an upperbound for the gain

achieved with either coordinated precoding or interference-aware precoding,

both of which operate on a frame-by-frame basis. Using full coordination

among the B bases, network MIMO requires both global CSIT and global

data and achieves a multiplexing gain per base of M .

5.4 Simulation methodology

In this section, we describe the metrics and methodologies used for numer-

ically evaluating the system-level performance of MIMO cellular networks.

Simulations are performed over a large network of hexagonal cells with B

bases and a total of KB users in the network. For a given network ar-

chitecture, users are dropped randomly and channel realizations are gen-

erated for many iterations. For a given iteration, the channel realizations are

fixed, and the performance for each base can be characterized by the K-

dimensional capacity region C obtained by treating the interference in (5.2)

and (5.4) as noise. To determine the optimal rate vector ROpt ∈ C, we use ei-

5.4 Simulation methodology 187

Interf.-awareprecoding

Precoding at base b is a function of the desired and

unintended users’channels

Some gains for cell edge users

Minimal improvement in

throughput

Base mode

Technique

Description

Benefits

Drawbacks

Multiplexinggain per

base

Base bknowledge

Coordinated precoding

Precoding at base b is a function of channels seen by

all coordinated bases

Some gains over interf.-aware

precoding

Requires global CSIT

Network MIMOprecoding

A cluster of coordinated bases transmit as a single base with distributed

antennas

Potentially optimal with full-network

coordination

Requires global CSIT and data, high complexity

Interf.-oblivious precoding

Precoding at base b is a

function of the desired user’s

channel

No exchange of base information

No explicit interference mitigation.

Interference alignment

Signals from interfering bases are aligned to lie in a “garbage”

subspace

Interference-free performance

Requires global CSIT,

high complexity.

Independent bases Coordinated bases

bases, antennas per base

users, antennas per user...

...

Fig. 5.16 Overview of spatial interference mitigation techniques for downlink cellular

networks. Multiplexing gain per base is given for a system with B bases, each serving one

user, and each base and user has M antennas.

ther proportional-fair or equal-rate performance criteria (described in Section

5.4.1) to achieve fairness among the K users.

These two methodologies will be used in the following chapter to evalu-

ate system performance of different MIMO techniques. The proportional-fair

methodology will be used primarily for evaluating the performance of inde-

pendent bases, and the equal-rate methodology will be used for evaluating

the performance of coordinated bases.

An iterative scheduling algorithm described in Section 5.4.2 can be used

to numerically calculate the optimum rate vectors under either criterion, and

it is also used in practice for scheduling in time-varying channels (Section

5.5.4). Rate vectors are collected over many iterations and across all bases

in the network to generate throughput and user-rate metrics. Details of the


proportional-fair and equal-rate simulation methodologies are given in Sec-

tions 5.4.3 and 5.4.4, respectively.

5.4.1 Performance criteria

In previous chapters, the sum-rate capacity is used for multiuser MIMO chan-

nels to determine the rate vector which maximizes the sum rate of the users.

For a base station serving K users with capacity region C, the sum-capacity

rate vector is given by

ROpt = argmaxR∈C

∑k∈U

Rk, (5.23)

where U is the set of K users assigned to this base. If the average SNRs of

the users are the same, then this criterion is fair in the sense that users would

receive the same average rate over many channel realizations. However, due

to co-channel interference in cellular networks, the geometry of the users can

vary significantly. The maximum sum-rate criterion would then not be fair

because maximizing the sum rate favors service to users with high geometry

while denying service to those with low geometry.

To provide more fairness, the proportional-fair criterion provides a higher

incentive to serve low-geometry users than sum-rate maximization. This is

achieved by maximizing the sum of log rates

ROpt = argmaxR∈C

∑k

logRk (5.24)

subject to the transmitter power constraint. It can be shown that an equiv-

alent condition for the proportional-fair vector ROpt in (5.24) is [148]

∑k

Rk −R∗k

R∗k

≤ 0 (5.25)

for any [R1, . . . , RK ] ∈ C. This means that in moving from the proportional-

fair vector ROpt to any other rate vector R ∈ C, the sum of the fractional

increases in user rates cannot be positive.

In the simulations for independent base operation in Chapter 6, each base

on the downlink is assumed to transmit with full power P and, on the uplink,

each user is assumed to transmit with full nominal power set according to


a power control algorithm. If there is only a single user per base (K = 1),

fairness is not a concern. In this case, each transmitter operates with full

power and the optimization in (5.24) becomes trivial.

As an alternative to the proportional-fair criterion, one could provide fair-

ness by maximizing the minimum rate achieved by any user:

ROpt = argmaxR∈C

mink∈U

Rk, (5.26)

where U is the set of users served by the base. The solution provides equal-

rate service to all users. The equal-rate criterion provides more fairness than

the proportional-fair criterion because there is no difference among the users’

rates. However, in cases with disparate SNRs, fairness could come at the

expense of lower throughput because the system could be forced to allocate

significant resources to the weakest user. As highlighted in Figure 5.17, the

goal of maximizing base station throughput is at odds with providing fairness

among users.

Performance criterion

Maximum sum rate

Proportional fair

Equal rate

Maximizes

Sum of rates

Sum of log rates

Minimum rate

Base station throughput

High

Medium

Low

Transmissionmode

Full-power

Full-power

Minimum sum-power

User fairness

High

Medium

Low

Fig. 5.17 Criteria for evaluating cellular network performance. Numerical simulations in

Chapter 6 will typically use the proportional-fair criterion for independent base operation

and the equal-rate criterion for coordinated base operation.

Figure 5.18 shows the two-user rate regions for a time-division multiple-

access (TDMA) channel, a multiple-access channel (with fixed covariances)

and a broadcast channel. In the TDMA channel, the base station is restricted

to serve one user at a time. Serving user 1 only, the rate vector (R∗1, 0) can

be achieved. Serving user 2 only, the rate vector (0, R∗2) can be achieved.

Other points on the rate region boundary can be achieved with time sharing.

If R∗1 > R∗

2, the sum-rate-maximizing rate vector is (R∗1, 0). The equal-rate

vector is the point (R1, R2) on the rate region boundary such that R1 = R2.

The proportional-fair vector, (R∗1/2, R

∗2/2), can be shown to satisfy (5.25) and

is achieved by serving each user half the time. It achieves a higher sum rate


than the equal-rate vector and achieves better fairness than the maximum-

sum-rate vector.

Sum-capacity plane

R1 (bps/Hz)

R 2(b

ps/H

z)

ox+

R1 (bps/Hz)

R 2(b

ps/H

z)

o

x

+

+ Equal-rateo Proportional-fairx Sum-rate maximizing

R1 (bps/Hz)

R 2(b

ps/H

z)+ o

TDMA MAC BC

Fig. 5.18 2-user capacity regions for TDMA, MAC (with fixed covariances), and BC

channels. Operating points for the three performance criteria are highlighted. For the

MAC, any vector lying on the sum-capacity plane maximizes the sum rate.

5.4.2 Iterative scheduling algorithm

Given the rate region C, we can find the sum-capacity rate vector (5.23)

using the numerical optimization techniques described in Section 3.5. The

proportional-fair rate vector (5.24) can be found by maximizing the sum of

log rates f(R) over C. Because this is a standard convex program, convex

optimization algorithms based on the ones in Section 3.5 can be used.

Alternatively, the proportional-fair vector can be found using an itera-

tive scheduling algorithm that maximizes the weighted sum rate during each

frame and then updates the quality of service (QoS) weights appropriately.

This algorithm is useful in practice for scheduling resources in time-varying

channels (Section 5.5.4). While we discuss this algorithm in the context

of time-invariant channels, a more general gradient scheduling algorithm is

asymptotically optimal in time-varying channels [149].

For a given set of channel realizations, the algorithm operates iteratively

over multiple frames. During frame f , the scheduler allocates resources to

achieve rate vector R(f) where


R(f) = argmaxR∈C

∑k

q(f)k Rk (5.27)

and the QoS weight q(f)k is a non-negative scalar. If we let the QoS weight

for the kth user be the reciprocal of the smoothed average rate:

q(f)k =

1

R(f)k

, (5.28)

where

R(f)k = αR

(f−1)k + (1− α)R

(f−1)k , (5.29)

and α ∈ [0, 1] is a forgetting factor, then the rate vector R(f) converges to the

proportional-fair vector for sufficiently small α and sufficiently large f [149].

We note that the rate vector that maximizes the weighted sum rate in (5.27)

can, in general, service multiple users if multiuser MIMO techniques are used.

If the transmission is restricted to serve a single user, then the solution to

the weighted sum rate problem is to serve the user with the largest weighted

rate. For single-user transmission, the achievable rate during frame f is

R(f) = argmaxR∈C

R(f)k

R(f)k

. (5.30)

The generalization of this procedure to time-varying channels is known as

the proportional-fair scheduling algorithm for single-user service [150].

In delay-intolerant applications such as streaming media, a minimum av-

erage rate could be required to meet QoS targets. The proportional-fair cri-

terion in (5.24) could be modified to ensure a minimum rate Rmin > 0 is

achieved by all users:

ROpt = argmaxR∈C,Rk≥Rmin∀k

∑k

logRk. (5.31)

The optimal ROpt satisfying (5.31) can be found using the same iterative

weighted sum-rate maximization in (5.27) but using a modified QoS weight

that depends on a token counter T(f)k [151]:

q(f)k =

exp(akT

(f)k

)R

(f)k

, (5.32)

where ak > 0 is an aggressiveness parameter. This token counter is updated

each frame by incrementing it with the minimum desired rate Rmin and re-


0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

Frame number

Rat

e (b

ps/H

z)

User 1 ratesaverageinstantaneous


Fig. 5.19 Instantaneous and average rates for a K = 2-user broadcast channel under

proportional-fair scheduling using DPC. The capacity region is the right subfigure of Figure

5.18. The initial rate vector achieved during frame 1 corresponds to the sum-rate capacity.

The steady-state rate vector is the proportional-fair rate vector which maximizes the sum

of log rates.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Frame number

Rat

e (b

ps/H

z)



Fig. 5.20 Instantaneous and average rates for a K = 2-user broadcast channel under

proportional-fair scheduling with minimum rate constraints. The constraints are set so

they are outside the achievable rate region. As a result, the rates converge to the equal-

rate vector.


0 5 10 15 20 25 30 35 400

5

10

15

20

25

Frame number

Rat

e (b

ps/H

z)

User 1User 2User 3User 4

Fig. 5.21 Instantaneous and average rates for a (4,4(1)) time-invariant broadcast channel

under proportional-fair scheduling and DPC. Instantaneous rates reach their steady-state

values after a few frames.


0 5 10 15 20 25 30 35 400

5

10

15

20

25

Frame number

Rat

e (b

ps/H

z)


Average rates:

Fig. 5.22 Instantaneous and average rates using ZF for the same (4,4(1)) channel used

in Figure 5.21. During steady-state operation, the transmitter serves users 1 and 2 during

even-numbered frames and users 3 and 4 during odd-numbered frames. Only half of the

M = 4 spatial degrees of freedom are used during each frame.


ducing it by the served rate:

T(f)k = max

(0, T

(f−1)k +Rmin −R

(f−1)k

). (5.33)

Using (5.32) in (5.27), the iterative algorithm converges to the optimal ROpt

satisfying (5.31) if that rate vector is achievable [149].

Figure 5.19 shows the instantaneous and average rates for 2-user broadcast

channel using DPC and proportional-fair scheduling. The capacity region of

this channel is shown in Figure 5.18. During the first frame, the QoS weights

for the two users are initialized to the same value. Since q1 = q2, then from

(5.27), the rate vector achieved during this frame maximizes the sum rate.

On subsequent frames, the QoS weights are updated according to (5.28) and

(5.29), and the average rates converge to the proportional-fair rates.

Figure 5.20 shows the instantaneous and average rates for the same 2-

user channel using DPC and proportional-fair scheduling with minimum rate

requirements. The minimum rate Rmin for both users is set to be outside the

achievable rate region. Because this rate vector is not achievable, the average

rates converge to the equal-rate vector.

Figure 5.21 shows the instantaneous and average rates for a time-invariant

broadcast channel with M = 4 antennas and K = 4 single-antenna users

using DPC. The sum-rate maximizing rate vector is achieved during frame

1 because the QoS weights are the same for all users. Even though there are

four spatial degrees of freedom, the sum rate is maximized by serving only two

users. When the performance reaches its steady state proportional-fair rate

vector, all users are served each frame. Figure 5.22 shows the scheduled rates

for the same channel using zero-forcing (Section 4.2.1). The users’ channels

are linearly independent so that up to four users could be served at once.

However, in steady state, only two users are served during each frame, and the

proportional-fair rate vector is achieved by time-multiplexing service between

two pairs of users. While user 1 achieves 20 bps/Hz whenever it is served, it

achieves this rate only half the time, so its average rate of 10 bps/Hz is below

that of its proportional-fair rate achieved under DPC (about 12 bps/Hz).

5.4.3 Methodology for proportional-fair criterion

Under the proportional-fair criterion for the downlink, bases transmit with

full power and user rates are adjusted so that the proportionally-fair rate


vector is achieved by each base. The same principle applies to the uplink,

except that users are power controlled to prevent excessive interference.

Figure 5.23 shows an overview of the proportional-fair methodology.

For a given iteration n, the location of users, shadowing realizations, and

channel fading realizations are fixed. An inner loop, running the iterative

scheduling algorithm described in Section 5.4.2 over many frames, provides

the proportional-fair rate vector (R(n)1,b , . . . , R

(n)K,b) for iteration n and base

b = 1, . . . , B. This methodology can be generalized for the case of time vary-

ing channels by allowing the channel fading to evolve from frame to frame.

5.4.3.1 Metrics

The performance metrics for the proportional-fair criterion are the mean

throughput per site, the peak user rate, and the cell-edge user rate. The

throughput measures the average data rate transmitted or received through

the S sectors of a cell site. The peak and cell-edge rates measure the perfor-

mance from the user’s perspective.

Proportional-fair rate vectors are collected across the bases and over multi-

ple iterations to generate the performance metrics. Each base serves K users,

and each base is associated with a sector. Therefore with B bases in the net-

work and S sectors per site, there are B/S sites in the network. Given the

user rates R(n)1,b , . . . , R

(n)K,b for bases b = 1, . . . , B and iterations n = 1, . . . , Q,

we define the mean throughput per site as

1

Q(B/S)

B∑b=1

K∑k=1

Q∑n=1

R(n)k,b = SK

1

KQB

B∑b=1

K∑k=1

Q∑n=1

R(n)k,b . (5.34)

The expression on the right shows that the mean throughput per site is the

mean user rate multiplied by the number of total users in the network SK.

By treating the user rates as realizations of a random variable R, we define

the peak user rate as the rate z such that the probability of R being less

than z is 90%. Using the inverse cumulative distribution function, the peak

user rate is F−1R (0.9). Similarly, the cell-edge user rate is the rate z such

that the probability of R being less than z is 10%: F−1R (0.1). Since the mean

throughput per site is SK times the mean user rate, we can make easier

comparisons with the cell-edge and peak user rates by normalizing these

values. We define the normalized peak user rate as SK × F−1R (0.9) and the

normalized cell-edge rate as SK × F−1R (0.1).


Initialize QoS weightsProportional-fair methodology

Determine rates for maximizing the weighted sum-rate metric

Rate vectors, iteration n

Update QoS weights

Place users and generatechannel realizations, iteration n

Assign users

Fig. 5.23 Overview of the proportional-fair simulation methodology. For iteration n, the

proportional-fair rate vector is found for the K users served by each base b = 1, . . . , B.(R1,b, . . . , RK,b

) ∈ Cb. The throughput and user rate metrics are computed from rate

vectors collected over many iterations.

5.4.3.2 Downlink

For a given network of B bases, we place a user indexed by k randomly

with a uniform spatial distribution in the network. With respect to each

base b = 1, . . . , B, we measure the distance dk,b to the user, determine the

directional antenna gain Gk,b, and generate a random shadowing realization

Zk,b. All bases are assumed to transmit with full power P , and the user

is assigned to the base with the highest SNR, or equivalently, the highest

channel gain (5.5). The serving base, which we denote with index b∗, is givenby

b∗ = argmaxb

α2k,b = argmax

b

(dk,bdref

)−γ

Zk,bGk,b, (5.35)

where the additional maximization with respect to the wraparound repli-

cas (Section 5.2.3) has not been explicitly stated. We repeat this process of


placing and assigning users until each base is assigned K users. If a user is

assigned to a base with K users already, then the user is simply discarded.

As explained in Section 1.2.2, the user geometry is an important metric for

characterizing the link performance in systems with co-channel interference.

The geometry is defined as the ratio between the average power received from

the desired source and the average noise plus interference power. From (5.4),

the downlink geometry for user k assigned to base b∗ is

Pα2k,b∗

σ2 +∑

b =b∗ Pα2k,b

=(P/σ2)α2

k,b∗

1 +∑

b =b∗(P/σ2)α2

k,b

, (5.36)

where the expression on the right gives the geometry in terms of the reference

SNR P/σ2. The geometry is a random variable that depends on the location

and shadowing realization of user k with respect to all the bases. As we

will see in Section 6.3, in a typical cellular network with universal reuse, the

geometry distribution has a typical range of -10 dB to 20 dB. If we treat

the interference as additive noise, then the relevant range of SNRs for the

equivalent isolated SU- or MU-MIMO channel is likewise -10 dB to 20 dB.

For a given placement of users and channel realizations, the proportional-

fair rate vectors for the B bases are determined using the iterative scheduling

algorithm from Section 5.4.2. Each step of this algorithm requires determin-

ing the rate vector that maximizes the weighted sum rate given the current

QoS weights. Given the received signal model (5.4), the transmit covariances

E(sbsHb ) of the interfering bases b �= b∗ are not known, and therefore it is not

possible to determine the optimal transmit covariance for base b∗ to maxi-

mize the weighted sum rate. Likewise, the transmit covariance for the other

bases cannot be determined without knowledge of their respective interfer-

ing bases. This chicken-and-egg problem is addressed by first computing the

transmit covariance of the serving base assuming the signal from the other

bases b �= b∗ is transmitted isotropically and averaged over the Rayleigh

fading. The covariance of the interference received from base b is

E(HH

k,bsbsHb Hk,b

)=

P

ME(HH

k,bHk,b

)= Pα2

k,bIN . (5.37)

Therefore we can rewrite the received signal (5.4) as

xk = HHk,b∗sb∗ + nk, (5.38)

where nk is the combined noise and interference modeled as a zero-mean

Gaussian vector with covariance


E(nkn

Hk

)= σ2IN + IN

∑b =b∗

α2k,bP. (5.39)

The transmit covarianceQb∗ can be calculated as if base b∗ were serving itsKusers in ZMSW noise with covariance (5.39). For example, for CL SU-MIMO

transmission, the transmit covariance for base b∗ is

Qb∗ = arg maxQ,trQ≤P

log2 det

[IN +

1

σ2 +∑

b =b∗ Pα2k,b

HHk,b∗QHk,b∗

]. (5.40)

The covariances of the other bases Qb, b = 1, . . . , B, b �= b∗ can be determined

in a similar manner.

Given the covariance Qb for base b, the interference received by user k

from base b, as a function of the channel realization Hk,b, has covariance

Hk,bQbHHk,b. The combined noise and interference in (5.4) has covariance

QI,k := σ2IN +∑b =b∗

HHk,bQbHk,b. (5.41)

Whereas the covariance in (5.39) is averaged over the channel realizations,

the covariance in (5.41) is a function of the block-fading channel realizations.

The interference covariance in (5.41) is assumed to be known at the re-

ceiver. Therefore the achievable rate for users served by base b∗ can be com-

puted based on the whitened received signal Q−1/2I,k xk, given the transmit

covariance Qb∗ determined in the presence of averaged interference. For the

example of SU transmission with covariance Qb∗ computed from (5.40), the

served mobile achieves rate

R = log2 det[IN +Q−1

I,kHHk,b∗Qb∗Hk,b∗

], (5.42)

where QI,k is from (5.41). The average rate (5.42), taken with respect to

realizations of the interferers’ channels (Hk,b, b �= b∗), can be lowered bounded

as follows:

EHk,b(R) ≥ log2 det

[IN +

1

σ2 +∑

b =b∗ Pα2k,b

HHb∗Qb∗Hb∗

]. (5.43)

This bound can be obtained by extending the derivation for (5.16) and ap-

plying Jensen’s inequality to the convex function

f(A) = log2 det(I+ (I+AHRA)−1X

). (5.44)


This inequality shows that the rate averaged with respect to the interference

with covariance from (5.41) will be no less than the rate achieved under the

average interference with covariance from (5.39).

In summary, the downlink transmit covariance for the proportional-fair

criterion is set under the assumption of averaged interference (5.39), but

the rate is computed using this transmit covariance and interference with

covariance (5.41).

5.4.3.3 Uplink

On the downlink, users are assigned to the base with the lowest pathloss.

As a consequence, because bases transmit with the same power, the signal

received by a user from its assigned base is always stronger than that from

any interfering base. On the uplink, users are also assigned to the base with

the lowest pathloss. However, as a result of shadowing, an interfering user

could be received at a base more strongly than its assigned user, resulting

in a very low geometry. Alternatively, the SINR could be very high if the

desired user is very close to its assigned base. In the interest of fairness, large

variations in geometry are not desirable, and power control is used to adjust

the transmit powers so that the geometry is near a nominal target ρ.

Due to power control, the transmit power for user k assigned to base b

(k ∈ Ub) is γkPk, where γk ∈ [0, 1] is the fraction of the power constraint

Pk. Fixing the power fraction of users assigned to other bases γj , j /∈ Ub, the

geometry of user k is

Γk(γk) =γkPkα

2k,b

σ2 +∑

j /∈UbγjPjα2

j,b

, (5.45)

where α2j,b is the channel gain (5.5) between user j and base b. The power

control algorithm adjusts the powers so that users with favorable channels

achieve the target geometry ρ transmitting power γkPk (with γk < 1), while

users with unfavorable channels achieve a geometry less than ρ even though

they transmit with full power Pk. For user k, the power fraction is set by fixing

the power fractions for other users and determining the required fraction γk

to achieve ρ or setting γk = 1 if ρ is not achievable:

γk =

⎧⎨⎩

ρ(σ2+

∑j /∈Ub

γjPjα2j,b

)

Pkα2k,b

if Γk(1) ≥ Pk

1 if Γk(1) < Pk.(5.46)


Power fractions are adjusted iteratively over all users until the values con-

verge.

If a user’s gain to its assigned base is much greater than the gain to the

other bases, it could potentially transmit with more power and increase its

geometry without causing significant interference. To account for this possi-

bility, we propose a more aggressive power control scheme which sets the new

target geometry to be

ρ

(α2k,b

α2k,b

)β

, (5.47)

where ρ is the nominal target, and α2k,b

is the channel gain between user k

and base b which has the next highest channel gain after its assigned base b.

The parameter β ≥ 0 adjusts the aggressiveness of the new target. Setting

β = 0 does not change the target geometry. A higher value of β gives a more

aggressive boost, resulting in a higher peak geometry but perhaps at the

expense of a lower tail.

Figure 5.24 shows the CDF of the uplink geometry (5.45) for the case

of universal reuse, S = 3 sectors per cell site, K = 1 user per sector, and

reference SNR P/σ2 = 30 dB. Using power control with a nominal target

SINR of ρ = 6 dB and an aggressiveness parameter β = 0.5, the range of

the geometry is significantly reduced compared to the case with no power

control where all users transmit with full power. In particular, the minimum

geometry with power control is about -14 dB, while it is less than -20 dB

without power control.

Once the power fractions γk are set by the power control algorithm off-

line, the uplink simulation methodology follows the downlink methodology

described above in Section 5.4.3.2, except of course that the multiple access

channel model is used instead of the broadcast channel.

The proportional-fair methodology, corresponding to the gray box in Fig-

ure 5.23, can be summarized as follows for both the downlink and uplink:

Proportional-fair algorithm

1. Initialize

Set f := 0. Set QoS weights q(f)k = 1/ε, (ε � 1) for all k = 1, . . . ,KB.

2. For f ≥ 0, do the following:

a. Update transmit covariances

Assuming average interference, determine transmit covariances to


maximize the weighted sum rate.

For the downlink, update base covariances Q(f)b , b = 1, . . . , B.

For the uplink, update user covariances Q(f)k , k = 1, . . . ,KB.

b. Determine achievable rate vector

In the presence of interference based on the covariances computed

in the previous step and using the instantaneous channel realiza-

tions, determine the rate vector for each base: R(f)1,b , . . . ,R

(f)K,b, for

b = 1, . . . , B.

c. Update QoS weights

Update q(f+1)k for k = 1, . . . ,KB, using, for example, (5.28).

d. Let f := f + 1 and iterate until convergence.

-20 -10 0 10 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Geometry (dB)

CD

F

Uplink,with power control

Uplink,no power control

Downlink

Fig. 5.24 CDF of uplink geometry (5.45) with and without power control. Power control

reduces the range of the upper and lower tails, making the distribution similar to the

downlink geometry distribution (Figure 6.15).


5.4.4 Methodology for equal-rate criterion

Under the equal-rate methodology, the performance is determined by the

achievable rate of the user with the poorest channel conditions. To prevent

the performance from being dominated by a few users with very poor channel

conditions, 10% of the users with the worst channels are declared to be in

outage and receive no service. Transmit powers are adjusted so that all users

in the network, except those in outage, receive a common data rate. Note

that under the proportional-fair methodology, users are not explicitly placed

in outage, and those with poor channels receive lower rates as a result of the

proportional-fair criterion.

The goal is to determine the largest common data rate that is consistent

with the target user outage probability of 10%. Figure 5.25 shows an overview

of the equal-rate methodology. For a given target rate R∗ and for a given iter-

ation f where the location of users and channel realizations are fixed, an inner

loop calculates the fraction of users u(f)R∗ in outage due to unfavorable channel

and/or interference conditions. The outage probability for R∗ is determined

by averaging u(f)R∗ over many iterations. The desired equal-rate performance

metric is the rate R∗ corresponding to the average outage probability of 10%.

While the equal-rate rate vector can be found using the iterative scheduling

algorithm (Section 5.4.2) with no users in outage, we describe a different

technique below that accounts for user outage. The technique determines the

outage probability for a few target rates R∗ and interpolates between these

rates to obtain the rate that results in 10% user outage. Below we describe the

details for the uplink and downlink channels where, for simplicity, we consider

linear transmitter and receiver processing and single-antenna terminals.

5.4.4.1 Uplink

We first present the uplink algorithm for the conventional network where

bases operate independently. We then present an extension for network

MIMO with coordinated bases.

Let the target rate for each user in the network be R∗ bps/Hz. Assuming

Gaussian signaling and ideal coding, the target rate of R∗ bps/Hz translates

to a target SINR of ρ � 2R∗ − 1 for each user. Suppose that the target SINR

ρ is small enough for all the users to achieve it, given the power constraint on

each user and the interference between users. This means that there exists a


Determine users in outage

Equal-rate methodology

Fix target rate R*

Place users and generatechannel realizations, iteration n

Assign users

Initialize powers

Users in outage, iteration n

Update powers to achieve rate R*

Fig. 5.25 Overview of equal-rate simulation methodology. For iteration n and for a given

target rate R∗, the outage probability is computed across users in the entire network.

Given R∗, the average outage probability is computed over many iterations. The target

rate corresponding to an average outage probability of 10% is determined by varying R∗.

feasible setting of each user’s transmitted power, and an assignment of users

to bases, such that each user attains an SINR of ρ or higher at its assigned

base, with an SINR-maximizing linear MMSE receiver.

If we denote the base assigned to user k as bk and the transmit power of

user k as pk, the SINR of user k is

pkhHk,bk

⎛⎝σ2IM +

∑j =k

pjhj,bkhHj,bk

⎞⎠−1

hk,bk . (5.48)

Note that this expression assumes perfect knowledge at base bk of the channel

vector hk,bk and the composite interference covariance∑

j =k pjhj,bkhHj,bk

. We

define an optimization problem for minimizing the total power as follows:

min{bk},{pk}

KB∑k=1

pk, (5.49)

subject to


pkhHk,bk

⎛⎝σ2IM +

∑j =k

pjhj,bkhHj,bk

⎞⎠−1

hk,bk ≥ ρ, pk ≥ 0. (5.50)

For this problem, the following iterative algorithm from [126] (also see [152,

153]) can be used to determine the transmitted powers and base assignments

for all the users. Letting p(f)k represent the uplink power of user k in the fth

frame, we first initialize all powers to zero: p(0)k = 0 for all k. For the fth

frame, given the set of powers{p(f)k

}, we assign each user k to its best base

and update its transmit power. The user is assigned to the base where it

would attain the highest SINR:

bk = arg maxb=1,...,B

hHk,b

⎛⎝σ2IM +

∑j =k

p(f)j hj,bh

Hj,b

⎞⎠−1

hk,b. (5.51)

The power for user k is updated for frame f + 1 so it achieves the target

SINR ρ at the assigned base bk, assuming every other user j transmits at the

current power level of p(f)j :

p(f+1)k = ρ

⎡⎢⎣hH

k,bk

⎛⎝σ2IM +

∑j =k

p(f)j hj,bh

Hj,b

⎞⎠−1

hk,bk

⎤⎥⎦−1

. (5.52)

The assignment and power updates are repeated for all other users, and the

entire procedure is repeated until convergence is achieved.

The algorithm above has been shown to converge [126] to transmitted

powers {pk} that not only minimize the sum powers in (5.49), but are optimal

in an even stronger sense: if it is possible for every user to attain the target

SINR of ρ with transmitted powers {pk}, then pk ≥ pk for every k. In other

words, the algorithm minimizes the power transmitted by every user, subject

to the target SINR of ρ being achieved by all users.

In general, it might be impossible for all the users to achieve the target

SINR simultaneously. It is then necessary to settle for serving only a subset

of the users, declaring the rest to be in outage. (This might be preferable

to serving all the users at a low rate determined by the user with the worst

channel conditions.) In principle, the largest supportable subset of users could

be determined by sequentially examining all subsets of users in decreasing

order of size, but this approach is practical only when the number of users is

small.


Instead, we will modify the iterative algorithm of [126] slightly to obtain

a suboptimal but computationally efficient algorithm for determining which

subset of users should be served. After each frame f , the modified algorithm

declares users whose updated powers exceed the power constraint of 1 to

be in outage, and eliminates them from consideration in future frames. This

progressive elimination of users eventually results in a subset of users that

can all simultaneously achieve the target SINR ρ. For this subset of users, the

algorithm then finds the optimal transmitted powers and cluster assignments.

However, the user subset itself need not be the largest possible; essentially,

this is because we do not allow a user consigned to outage to be resurrected

in a future frame.

The algorithm described above applies to a conventional cellular network

where bases operate independently. It can be extended to the case of uplink

network MIMO where clusters of bases work jointly to demodulate a user’s

signal. A coordination cluster is defined to be a subset of base stations that

jointly serve one or more users through the antennas of all their sectors. The

network is postulated to have a predefined set of coordination clusters, and

each user can be assigned to any one of these clusters.

To highlight the dependence of the spectral efficiency gain on the number

of coordinated bases, coordination clusters with a specific structure are of

interest. For any integer r ≥ 0, an r-ring coordination cluster is defined to

consist of any base station and the first r rings of its neighboring base stations

(accounting for wraparound), and Cr is defined to be the set of all r-ring

coordination clusters in the network. Figure 5.26 illustrates r-ring clusters

for r = 0, 1, 2, 4.

With C0 as the set of coordination clusters in the network, there is in

fact no coordination between base stations. This case serves as the bench-

mark in estimating the spectral efficiency gain achievable with sets of larger

coordination clusters.

With some abuse of notation, let hk,C ∈ C3N |C| denote the channel from

user k to the antennas of all the base stations in the coordination clus-

ter C; here |C| denotes the number of base stations in C, and the fac-

tor of 3 comes from assuming S = 3 sectors per site. Then, with user

k transmitting power pk, the SINR attained by user k at cluster C is

h∗k,C

(σ2IM +

∑j =k pjhj,Ch

∗j,C

)−1

hk,C pk. Using this expression, we can

generalize the original algorithm for conventional networks to the case of net-

work MIMO in a straightforward manner by replacing the user assignment


s

Fig. 5.26 Coordination clusters with r rings. The case of r = 0 denotes intrasite coordi-

nation where the sector antennas at a site are coordinated.

(5.51) with

C(f)k = arg max

C∈Cr

hHk,C

⎛⎝σ2IM +

∑j =k

p(f)j hj,Ch

Hj,C

⎞⎠−1

hk,C , (5.53)

and by replacing the power update (5.52) with

p(f+1)k = ρ

⎡⎢⎣hH

k,C(f)k

⎛⎝σ2IM +

∑j =k

p(f)j h

j,C(f)k

hH

j,C(f)k

⎞⎠−1

hk,C

(f)k

⎤⎥⎦−1

. (5.54)

The uplink equal-rate methodology, corresponding to the gray box in Figure

5.25, is summarized below.

Uplink equal-rate algorithm

1. Initialize

Set f := 0, start with no users in outage, and let p(f)k = 0 for all

k = 1, . . . ,KB.

2. For frame f ≥ 0, given{p(f)k

}, do the following:

a. Assign user

Assign each user (k = 1, . . . ,KB) to the base/cluster where it

would attain the highest SINR: (5.51) for a conventional network

or (5.53) for network MIMO.


b. Update transmit power

Update for each user (k = 1, . . . ,KB) the uplink power p(f+1)k :

(5.52) for conventional or (5.54) for network MIMO.

c. Determine users in outage

For each base (b = 1, . . . , B) and for each user k ∈ Ub, if

p(f+1)k > 1, eliminate the user and reset its power to zero in all

future frames.

d. Let f := f + 1 and iterate until convergence.

5.4.4.2 Downlink

As in the uplink, all users in the downlink equal-rate methodology, except

those in outage, must be served at a common data rate R∗ with correspond-

ing SINR ρ. As before, our goal is to determine the largest common rate that

is consistent with the desired user outage probability. However, unlike the up-

link where the interference covariance and CSI of all users is known at a given

receiving base—this assumption is reasonable since the covariances could be

estimated from the interfering users’ pilots signals—the downlink CSI of all

users is not necessarily known at any given base. We consider the following

three transceiver architectures: single-base precoding with local CSIT, co-

ordinated precoding with global CSIT, and network MIMO precoding with

global CSIT and global data.

Under the assumption of linear precoding, the transmitted signal from

base b can be written as

sb =∑j∈Ub

√vj,bgj,buj , (5.55)

where Ub is the set of users served by base b, vj,b is the transmit power

allocated by base b to user j, gj,b ∈ CM is the corresponding normalized

beamforming vector with unit norm, and uj is the unit-power information-

bearing signal for user j. We will denote by bk the base station to which

user k is assigned. Using (5.55) in (5.4), the received signal for user k under

conventional network operation can then be written as

xk = hHk,bk

sbk +∑b =bk

hHk,bsb + nk (5.56)


=√vk,bkh

Hk,bk

gk,bkuk +∑

j∈Ubk,j =k

√vj,bkh

Hk,bk

gj,bkuj + (5.57)

∑b =bk

∑j∈Ub

√vj,bh

Hk,bgj,buj + nk. (5.58)

Given the beamforming vectors for all users, the SINR of user k assigned to

base bk is|hH

k,bkgk,bk |2vk,bk

σ2 +∑

j∈Ubk,j =k |hH

k,bkgj,bk |2vj,bk + μk

, (5.59)

where

μk =∑b =bk

∑j∈Ub

vj,b|hHk,bgj,b|2 (5.60)

is the intercell interference received from the other bases. The downlink op-

timization problem for the conventional network can be stated as follows:

min{bk},{gk},{vk}

KB∑k=1

vk (5.61)

subject to

|hHk,bk

gk,bk |2vk,bkσ2 +

∑j =k,j∈Ubk

|hHk,bk

gj,bk |2vj,bk + μk≥ ρ, (5.62)

||gk,bk || = 1, (5.63)

vk,bk ≥ 0, (5.64)

for k = 1, . . . ,KB, where μk is given by (5.60). In contrast to the uplink op-

timization where the SINR (5.48) could be expressed implicitly as a function

of the MMSE receiver, the downlink SINR (5.59) requires the beamforming

vector to be given explicitly.

To solve this optimization we use the linear BC/MAC duality from Sec-

tion 4.3: if a given rate vector is achievable in one direction for given sets

of base assignments and beamforming weights, then it is achievable in the

other direction with the same base assignments, same beamforming weights,

and same sum power. We could therefore run the dual uplink algorithm until

convergence. Then, fixing the base assignments and beamforming weights,

iterate until downlink powers converge. In theory this procedure would re-

quire dual infinite frame iterations. A more elegant algorithm based on [124]

would run the frames in parallel so that the downlink powers {q(f)k } during

frame f are updated immediately after the fth frame of the algorithm for


determining the dual uplink powers {p(f)k }. The parallel algorithm can be

shown to converge to the solution that minimizes the sum of all transmitted

powers. In contrast to the uplink, however, it is in general not possible to

minimize the individual user powers. Normalizing the channel so the total

interference plus noise power is unity, the downlink received signal (5.56) for

user k = 1, . . . ,KB can be written as

xk =1√

σ2 + μk

hHk,bsb + n′

k, (5.65)

where n′k has unit variance. The dual uplink received signal at base b =

1, . . . , B is therefore:

xb =∑k∈Ub

1√σ2 + μk

hk,bsk + n′b, (5.66)

where n′b has unit variance. The downlink algorithm iteratively updates the

user assignment, the MMSE beamforming vector for the dual uplink, the

downlink interference power, the dual uplink transmit powers, and the down-

link transmit powers.

The downlink algorithm is initialized with no users in outage and with

q(0)k = p

(0)k = 0 for all k. The interference is initialized to μ(−1) = 0. During

frame f , user k is assigned to the base where it would attain the highest dual

uplink SINR (5.59):

b(f)k = arg max

b=1,...,BhHk,b

⎛⎝IM +

∑j∈Ub,j =k

p(f)j

σ2 + μ(f−1)j

hj,bhHj,b

⎞⎠−1

hk,b. (5.67)

For each user k ∈ Ub, the corresponding unit-norm beamforming weight

vector g(f)k derived from the MMSE criterion for the dual uplink is:

g(f)k =

(IN +

∑j∈Ub,j =k

p(f)j

σ2+μ(f−1)j

hj,bhHj,b

)−1

hk,b∥∥∥∥∥(IN +

∑j∈Ub,j =k

p(f)j

σ2+μ(f−1)j

hj,bhHj,b

)−1

hk,b

∥∥∥∥∥. (5.68)

To simplify the notation, we have dropped the user and frame indices for the

assigned base b(f)k . For each base b = 1, . . . , B, and for each user k ∈ Ub, the

interference power received from other bases b′ �= b is updated:


μ(f)k =

∑b′ =b

∑j∈Ub′

q(f)j

∣∣∣hHk,b′g

(f)j

∣∣∣2 . (5.69)

Using the dual uplink received signal (5.66), the uplink power for user k ∈ Ub

is updated so that it achieves the target SINR ρ:

p(f+1)k =

ρ

(1 +∑

j =kj∈Ub

p(f)j

σ2+μ(f−1)j

∣∣∣hHj,bg

(f)k

∣∣∣2)∣∣∣hH

k,bg(f)k

∣∣∣2

σ2+μ(f−1)k

(5.70)

Using the downlink SINR expression in (5.59), the downlink transmit power

is updated for each user so that it achieves the target SINR ρ. For each base

b = 1, . . . , B, and for each k ∈ Ub, the power is set according to:

v(f+1)k,b =

ρ(1 +∑

j∈Ub,j =k |hHk,bgj |2v(f)j,b + μk

)|hH

k,bg(f)k |2

. (5.71)

The algorithm is repeated over many frames until convergence occurs. As

was done for the uplink in the case of outage, we can modify the downlink

algorithm to obtain a suboptimal but computationally efficient algorithm

for determining the subset of served users. The basic idea is to check, after

each frame, whether the power constraint at any sector antenna is violated

and, if so, to eliminate users in decreasing order of requested power from

that antenna until all power constraints are met. This approach is subopti-

mal because a user cannot be resurrected once assigned to the outage state.

However, we use this technique because the optimal approach is prohibitively

complex for typical network sizes.

We now discuss the related methodology for the case for interference

nulling and network MIMO where CSI is known at all base stations. We de-

scribe the methodology for the more general case of network MIMO because

interference nulling is a special case when each user’s coordination cluster

contains a single base. In general, we let Ck denote the set of bases belong-

ing to the coordination cluster that serves user k. Note that given Ck for all

k = 1, . . . ,KB, we can determine Ub, the set of users served by base b, for

all b = 1, .., B, and vice versa. Under network MIMO, any base belonging to

cluster Ck has knowledge of that user’s data.

Given the coordination clusters Ck for all users k = 1, . . . ,KB, the received

signal for a user k (5.3) can be written as


xk =√vkhk,Ck

gkuk +∑j =k

√vjhk,Cj gjuj + nk, (5.72)

where hk,Cj ∈ CM |Cj | is the vector of stacked channels from bases serving

user j to the user k and where gj ∈ CM |Cj | is the unit-norm vector of stacked

beamforming vectors for the bases in the set Cj .

Given the beamforming vectors for all users in (5.72), the SINR of user k

is ∣∣∣hHk,Ck

gk

∣∣∣2 vkσ2 +

∑j =k

∣∣∣hHk,Cj

gj

∣∣∣2 vj . (5.73)

The downlink optimization problem under network MIMO can be stated as

follows:

min{Ck},{gk},{vk}

KB∑k=1

vk (5.74)

subject to ∣∣∣hHk,Ck

gk

∣∣∣2 vkσ2 +

∑j =k

∣∣∣hHk,Cj

gj

∣∣∣2 vj ≥ ρ, ‖gk‖ = 1, vk ≥ 0. (5.75)

The iterative algorithm is similar to the previous downlink algorithm for the

conventional network except that the channels do not need to be normalized

by the noise plus interference power.

During frame f , user k is assigned to the base where it would attain the

highest dual uplink SINR:

C(f)k = arg max

C∈Cr

hHk,C

⎛⎝σ2IM +

∑j =k

p(f)j hj,Ch

Hj,C

⎞⎠−1

hk,C , (5.76)

where the summation is taken over users j not in outage. The corresponding

unit-norm beamforming weights g(f)k derived from the MMSE criterion for

the dual uplink is

g(f)k =

(σ2IN +

∑j =k p

(f)j h

j,C(f)k

hH

j,C(f)k

)−1

hk,C

(f)k∥∥∥∥∥

(σ2IN +

∑j =k p

(f)j h

j,C(f)k

h∗j,C

(f)k

)−1

hk,C

(f)k

∥∥∥∥∥. (5.77)


Update the dual uplink power p(f+1)k and downlink power q

(f+1)k for user k

to attain the target SINR ρ assuming all other powers are fixed:

p(f+1)k =

ρ

[σ2 +

∑j =k p

(f)j

∣∣∣∣hH

j,C(f)k

g(f)k

∣∣∣∣2]

∣∣∣∣hH

k,C(f)k

g(f)k

∣∣∣∣2(5.78)

v(f+1)k =

ρ

[σ2 +

∑j =k v

(f)j

∣∣∣∣hH

k,C(f)j

g(f)j

∣∣∣∣2]

∣∣∣∣hH

k,C(f)k

g(f)k

∣∣∣∣2. (5.79)

The downlink equal-rate methodology, corresponding to the gray box in

Figure 5.25, is summarized below.

Downlink equal-rate algorithm

1. Initialize

Set f := 0, start with no users in outage. Let p(f)k = v

(0)k = 0 for all

k = 1, . . . ,KB. For conventional networks, let μ(f−1)k = 0.

2. For f ≥ 0, given{p(f)k

}and

{v(f)k

}(and

{μ(f−1)k

}), do the following:

a. Assign users

Assign each user (k = 1, . . . ,KB) to the base/cluster where it

would attain the highest dual uplink SINR: (5.67) for conventional

or (5.76) for network MIMO.

b. Compute dual uplink beamformers

Compute for each user (k = 1, . . . ,KB) the corresponding unit-

norm beamforming weight derived from the MMSE criterion for

the dual uplink: (5.68) for conventional or (5.77) for network

MIMO.

For conventional networks, update the interference power (5.69).

c. Update transmit powers

Update for each user (k = 1, . . . ,KB) the dual uplink powers

p(f+1)k :


Update for each user (k = 1, . . . ,KB) the downlink powers v(f+1)k :



d. Determine users in outage

For each base b = 1, . . . , B, if∑

k∈Ubv(f+1)k,b > 1, then eliminate

users in decreasing order of requested power until the power con-

straint is met.

e. Let f := f + 1 and iterate until convergence.


In this section, we discuss some practical aspects of implementing a cellu-

lar network, including the antenna array design, resource scheduling, and

challenges for base station coordination.

5.5.1 Antenna array architectures

The antenna array architecture for a given sector is characterized by the num-

ber of antenna elements, the spatial correlation between the elements, and

their directional response Gk,b of the sector beam pattern. More sophisticated

models account for the vertical beam pattern as well.

Figure 5.27 shows the physical manifestations of different array architec-

tures used for each sector of a site with S = 3 sectors. Each rectangle denotes

a radome enclosure and contains a column of either vertically polarized (V-

pol) elements or cross-polarized (X-pol) elements. Within a V-pol column,

the elements are co-phased to provide vertical aperture gain. Typically, the

3 dB beamwidth measured in the vertical direction is less than 10 degrees.

Because the beamwidth is inversely proportional to the aperture size [154]

and because the desired vertical beamwidth is narrower than the horizontal

beamwidth (about 65 degrees for the case of 3-sector sites), the resulting an-

tenna is longer in the vertical dimension. To help shape the beam pattern,

a physical reflector is placed behind the array of elements to direct energy

in the broadside direction. Note that the 1V configuration consists of mul-

tiple co-phased elements but for the purposes of performance evaluation is

modeled by a single M = 1 antenna.


A commercial 1V antenna designed for a 2 GHz carrier frequency and S =

3 sectors per site (whose response is shown in Figure 5.6) has a height, width,

and depth of, respectively, 130 cm, 15 cm, and 7 cm. The 3 dB beamwidth in

the vertical and horizontal dimensions are 7 and 65 degrees, respectively. The

dimensions are roughly proportional to the carrier wavelength, so an antenna

with the same characteristics operating at 1 GHz would be twice as large in

each dimension.

An X-pol antenna in Figure 5.27 consists of a column of dual-slant elements

with ±45 degree polarizations. The elements with different polarizations are

orthogonal, so the X-pol antenna is modeled by M = 2 spatially uncorrelated

antennas. The two antennas provide diversity, so the configuration is known

as DIV-1X. Compared to the 1V antenna, the DIV-1X antenna is slightly

wider. Diversity could also be achieved using two V-pol columns that have

sufficient spatial separation (Section 2.4).

Base station antennas with half-wavelength spacing are highly correlated

and can be used to form directional beams. The ULA-2V and ULA-4V config-

urations consist of, respectively, two and four columns of vertically polarized

elements arranged in a uniform linear array (ULA) with half-wavelength spac-

ing between columns. Because the aperture of ULA-4V is about twice that

of the ULA-2V, it can form beams with about half the beamwidth.

The CLA-2X configuration consists of two X-pol columns with close (half-

wavelength) spacing. One beam can be formed using the two columns of +45

degree polarization, and another beam can be formed in the same direction

using the two columns of -45 degree polarization. Because the beams are

formed using different polarizations, they will be uncorrelated. Therefore the

CLA-2X configuration gives the benefits of both direction beamforming and

diversity.

The DIV-2X configuration consists of two widely spaced columns to pro-

vide diversity. The spacing of the columns relative to the angle spread is

sufficient so the columns are spatially uncorrelated. The DIV-2X configura-

tion therefore achieves diversity order four. An array consisting of four V-pol

columns could achieve the same diversity, but the array would be larger.

As discussed in Chapter 7, next-generation standards such as LTE-Advanced

and IEEE 802.16m support MIMO techniques that use up to M = 8 anten-

nas. Possible architectures include four closely-spaced X-pol columns, four

widely spaced X-pol columns, or eight closely-space V-pol columns.


Antennas perconfiguration

Single-columnV-pol

Antennaconfiguration ULA-2V DIV-1X TX-DIV ULA-4V CLA-2X DIV-2X

Closely spacedV-pol

columns

Single-columnX-pol

2 41

1V

Description

Diversity spacedV-pol

columns

Closely spacedV-pol

columns

Closely spacedX-pol

columns

DiversityspacedX-pol

columns

Physicalform

Fig. 5.27 Examples of antenna array architectures. Antenna elements in each column are

co-phased to form a single virtual element.

5.5.2 Sectorization

For a site with S = 3 sectors and M = 1 antenna per sector, a 1V antenna

can be used to create each sector. Using a ULA-2V array whose width is

twice that of the 1V antenna, fixed beamforming weights implemented in the

RF hardware can be applied to the two columns to create a beam whose

width is half that of the 1V antenna’s. Each ULA-2V array is enclosed in

a single radome and is known as a sector antenna, and six ULA-2V arrays

could be used to support a site with S = 6 sectors. Even though two columns

of elements are used to create the sector, the weights across the elements are

fixed, and there is effectively M = 1 antenna per sector. Similarly, twelve

sector antennas, each implemented as a ULA-4V array, could support a site

with S = 12 sectors and M = 1 antenna per sector.

Figure 5.28 shows the overhead view of different antenna configurations

for cell sites. Columns A, B, and C show the site antenna configurations for

S = 3, 6, and 12 sectors implemented with sector antennas. In doubling the

number of sectors from S = 3 to 6 using sector antennas, there are twice

as many antennas and each antenna doubles in width. Therefore the total

weight of the antennas increases roughly by a factor of 4. The total weight


again increases by a factor of 4 in going from S = 6 to 12 sectors. Larger

antennas require additional infrastructure to cope with the higher weight and

wind load, and they also are visually more obtrusive.

To support multiple M > 1 antennas per sector, the service provider could

deploy M sector antennas in each sector. For example in column D of Figure

5.28, a pair of diversity-spaced V-pol columns (TX-DIV) is used for each

sector of a S = 3-sector site to implement M = 2 uncorrelated antennas.

Alternatively, in column E, a DIV-1X antenna could be used to implement

M = 2 uncorrelated antennas with a smaller array footprint.

A linear array with closely spaced elements can be used as a sector antenna

to form a single directional beam. It could also be used to form multiple si-

multaneous beams by weighting signals differently across the elements. The

weighting could be performed in the RF domain through hardware (for ex-

ample with Butler matrices) or electronically in the baseband domain. This

implementation is sometimes known as a multibeam antenna. For example, as

shown in column F of Figure 5.28, a ULA-2V array could create two beams,

one pointing 30 degrees with respect to the broadside direction and another

pointing at -30 degrees. Because the beams are not in the broadside direction,

the sidelobe and beamwidth characteristics will not be as good as the one

formed under column B’s configuration. However, the antenna array foot-

print will be smaller. Alternatively, the ULA-2V configuration could be used

to provide M = 2 correlated antennas for a sector of a S = 3-sector site.

Column G of Figure 5.28 shows a site with three ULA-4V arrays. Each array

could implement M = 4 correlated antennas in each of S = 3 sectors. Or each

could create four fixed directional beams to support a total of S = 12 sec-

tors. The characteristics of the directional beams can be improved by using

additional elements, for example, using eight closely spaced V-pol columns

instead of four. But the tradeoff is that the array is twice as wide. The beam

characteristics and performance of these different antenna architectures will

be discussed in Section 6.4.

Beams implemented in the RF hardware are fixed. Electronically generated

beams provide more flexibility because they can be dynamically adjusted

by changing the baseband beamforming weights. In channels with low angle

spread, these dynamic beams could track the direction of mobiles using CDIT

precoding (Section 4.2.2). The disadvantage of electronically generated beams

is that the antenna elements need to be phase calibrated.

As discussed in Section 5.2.1, the effective sector response is widened as a

result of channels with non-zero angle spread. Because the intersector inter-


ference increases as the sector response becomes wider, higher-order sector-

ization is most effective if the angle spread is significantly smaller than the

sector width. The angle spread associated with a base station tower decreases

as the height of the tower increases relative to the height of the surrounding

scatterers. For a fixed tower height, increasing the sectorization order will

result in diminishing returns on throughput performance as the intersector

interference overwhelms the multiplexing gains. In this regime, increasing

the tower height could result in a lower angle spread and reduced intersector

interference.

Angle spread distributions of a typical suburban channel and two urban

macrocellular channels (derived from [36]) are shown in Figure 5.29. The

suburban angle spread is less than 10 degrees with nearly probability 1. The

angle spread of the urban channels is higher but is still less than 20 degrees

with a significant probability. In all cases, dispersion due to angle spread

would be negligible for S = 3 sectors. Suburban channels could support

S = 12 channels without significant dispersion and urban channels could

support fewer.

A B C D E F

ULA-1V

3

1

Overheadview

Antenna configuration

Sectors per cell site (S)

Antennas per sector (M)

ULA-2V

6

1

ULA-4V

12

1

DIV-1X

3

2UC

TX-DIV

3

2UC

ULA-2V

3

2C

6

1

ULA-4V

3

4C

12

1

G

UC:C:

spatially uncorrelated antennasspatially correlated antennas

Fig. 5.28 Overhead view of a base station site showing antenna array cross-sections. The

different antenna configurations are shown in Figure 5.27.


suburban

urban

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Angle spread (degrees)

CD

F

Fig. 5.29 Angle spread distribution of typical suburban and urban macrocellular channels

[36]. The mean angle spread of the suburban and urban channels are, respectively, 5 and

17 degrees.

5.5.3 Signaling to support MIMO

In wireless systems such as cellular networks, auxiliary signaling is used to

support MIMO transmission. For example, we have discussed in Sections

2.3.7 and 4.2.3 the feedback bits used to indicate a user’s preferred codeword

selected from the codebook B. These B feedback bits are known as precoding

matrix indicator (PMI) bits in the context of 3GPP LTE standards. Because

the codewords are assumed to have unit power, the PMI conveys only direc-

tional information about the measured channel.

In order to set the coding and modulation for a data stream, an estimate

of the channel magnitude, as measured by the SINR or geometry, is fed back

from the mobile to the base. Due to the limited bandwidth on the uplink

feedback channel, this estimate is quantized to a few bits. In 3GPP standards,

they are known as channel quality indicator (CQI) bits. For a slowly fading

SISO channel, reference signals associated with each base antenna can be

used to estimate the channel hb∗ with respect to the serving base b∗ and the


channels hb, b �= b∗, of the interfering bases (Section 4.4.2). In this case, the

CQI is a quantized measurement of the SINR

|hb∗ |2 Pb∗

σ2 +∑

b=b∗ |hb|2 Pb

. (5.80)

For MISO channels, common reference signals allow the estimation of the

channels hHb∗ and hH

b , b �= b∗ (Section 4.4.2). If the desired base uses precoding

vector gb∗ , and the interfering bases use vectors gb, b �= b∗, the SINR would

be ∣∣hHb∗gb∗

∣∣2 Pb∗

σ2 +∑

b=b∗∣∣hH

b gb

∣∣2 Pb

. (5.81)

However, since the precoding vectors of the interfering bases are not known,

the user can approximate the interference power by assuming each base trans-

mits isotropically. Then the CQI would be based on∣∣hHb∗gb∗

∣∣2 Pb∗

σ2 +∑

b=b∗ ‖hb‖2 Pb

. (5.82)

For SU-MIMO transmission, spatially multiplexed streams could be en-

coded and modulated independently. In this case, CQI is fed back for each

stream separately.

5.5.4 Scheduling for packet-switched networks

As described briefly in Chapter 1, a scheduler is used in packet-switched net-

works for dynamically allocating spectral resources among a base’s assigned

users. A block diagram of the scheduler and data transmission is shown in

Figure 5.30. Scheduling is performed at each base on a frame-by-frame basis

and is done independently for the uplink and downlink. Users are said to be

active if they are scheduled to transmit data on the uplink or receive data

on the downlink. The scheduler determines which users are active and how

the frequency resources are allocated. If the base has M > 1 antennas, it de-

termines how the spatial resources are allocated for spatial multiplexing. For

each active user, the scheduler determines the precoding matrix, the number

of streams transmitted (given by the rank of the precoding matrix), and the

modulation and channel coding scheme for each stream.


For each user k = 1, . . . ,K, the scheduler requires the following informa-

tion:

• performance metrics, such as the average achieved rate,

• QoS requirements, such as the maximum tolerated latency or minimum

required rate,

• data statistics, such as the data buffer size,

• CSIT or PMI, if applicable, for precoding,

• CQI for determining the modulation and coding scheme.

The performance metrics, QoS requirements, and data statistics are readily

available at the base. The PMI and CQI information is obtained through

feedback from the mobiles, as described below in Section 5.5.5. Over a period

of many frames, the scheduler seeks to optimize long-term objectives such

as maximizing the mean throughput subject to user-specific rate and delay

requirements. It can do so by appropriately updating the QoS weights as

described in Section 5.4.2 and allocating resources to maximize the short-

term weighted sum rate. We note that the base station ultimately decides how

resources are allocated, and it is not obliged to follow the recommendations

suggested by the CQI and PMI feedback. For example, even though a mobile

feeds back a CQI to indicate it has a very high SINR, the base may schedule

it a low rate transmission because there is very little data in the buffer for

that user.

We have focused so far on the case of static, time-invariant channels.

In practice, the channels experience time variations due to fading, and the

channel-aware scheduler can exploit the variations. Figure 5.31 shows ex-

amples of two scheduling algorithms for a narrowband (1, (2, 1)) broadcast

channel where a single-antenna base serves two single-antenna users over in-

dependently fading channels and where one active user is served per frame.

As a result of the Rayleigh fading, the SINR of each user varies over time.

The frame-by-frame SINR measured by each user is plotted versus time, and

the scheduled user for the two algorithms is shown at the bottom of the fig-

ure. Round-robin scheduling is a simple algorithm which does not use the

channel measurements. Each user is served on alternating frames regardless

of the channel realizations. The channel-aware scheduler employs a simpli-

fied strategy for scheduling the user whose measured SINR is better than its

average SINR. Over many frames, each user will be served half the time on

average, and this strategy achieves a higher rate for each user than round-

robin scheduling. As the number of users increases, the probability of a user


having a very good fade during any frame increases, and the advantage of

the channel-aware strategy is even higher compared to round-robin.

This effect of achieving improved system performance by exploiting in-

dependent fading across multiple users is known as multiuser diversity

[121, 155, 156]. In multiuser diversity, fading across users is exploited in or-

der to improve the system performance, and larger fading variations actually

provide larger performance gains for multiuser diversity. Because transmit

diversity reduces the signal variations, it can actually result in decreased

throughput compared to a system with no diversity, when channel-aware

schedulers are employed. In fact, in the limit of a large number of users and

antennas, the maximum throughput achieved by any optimal scheduling al-

gorithm can be infinitely worse than a system with no diversity [157].

Resource allocation:Maximize weighted

sum of rates.

CQI, CSIT/PMI

QoS weight computationQoS weights:

Performance metrics

QoS requirementsData statistics

Data transmission

Data detectionChannel estimationAcknowledgement

Update performance metrics

Active usersPrecoding weightsModulation and coding

SchedulerFor users 1,…, :

Fig. 5.30 Overview of scheduling and data transmission in a packet-switched cellular

network with scheduling. On a frame-by-frame basis, each base schedules and serves its

assigned users 1, . . . ,K. Link adaptation is used to adjust rates based on the user’s QoS

and channel information. The acquisition of CQI, CSIT, and PMI is described in Figure

5.32.


User 1 average SINR

User 2 average SINR

Round-robin schedulingChannel-aware scheduling

1 2 1 2 1 2 1 2

1 1 2 2 2 1 1 2

User 1 measured SINR

User 2 measured SINR

Time

SINR

Scheduled user

Fig. 5.31 Round-robin and channel-aware scheduling for a single-antenna base serving

K = 2 users. The channel is assumed to be block fading.

5.5.5 Acquiring CQI, CSIT, and PMI

The steps for acquiring the CQI and CSIT/PMI information for scheduling

are shown in Figure 5.32.

For the uplink, the base estimates the CSI of each active user based on

their pilot signals and also the SINR using pilots from users assigned to

other bases. Using the CSI and SINR estimates with the other user statistics

available at the base, the scheduler determines the active users, data rates,

and precoding matrices, if applicable. This information is conveyed to the

users on a downlink control channel, and the active users transmit their data.

For the downlink, the sequence of events depends on whether the system

is time-division duplexed (TDD) or frequency-division duplexed (FDD). In

TDD systems, uplink and downlink transmissions occur over the same band-

width but are duplexed in time. CSI estimates obtained from uplink user

pilots could be used on the downlink, and the CSIT would be reliable if the

channel changes slowly relative to the time-duplexing interval, as discussed in

Section 4.4. The SINR of each user cannot be estimated from uplink trans-

missions and must be measured at the user. This information is conveyed

back to the base on an uplink feedback control channel. With all the neces-

sary information in hand, the base makes a scheduling decision and transmits

to its active users.


In FDD systems, the SINR is estimated at the mobile and fed back to the

base. CSI or PMI are likewise fed back on the uplink. For CDIT precoding, if

the uplink and downlink channels are statistically correlated, the base could

estimate the UL CDI and use this information for the downlink precoding.

The iterative scheduling algorithm described in Section 5.4.2 was described

for time-invariant channels. The algorithm can be applied to time-varying

channels by maximizing the weighted sum rate (5.27) using a rate vector

R lying in the rate region C(f) that varies as a function of the time index

f . This algorithm forms the basis of channel-aware scheduling and can be

extended for time-frequency resource allocation in wideband OFDMA sys-

tems [158]. The more general greedy-prime-dual algorithm [151] can also be

used to dynamically allocate frequency resources in OFDMA systems [159].

Uplink

Base estimates UL SINR and CSI

Base makes scheduling decision

Base informs active users

Active users transmit data

Downlink TDD

Base estimates UL CSI

Users estimate DL SINR

Users feed back CQI


Base transmits data to active users

Downlink FDD

Users estimate DL SINR and CSI

Users feed back CQI and PMI


Base transmits data to active users

Fig. 5.32 Steps for acquiring CQI, CSI, and PMI information.

5.5.6 Coordinated base stations

Figure 5.33 shows architectures for implementing coordination base station

techniques. Uplink network MIMO can be implemented using a centralized

architecture as shown in the top subfigure for a coordination cluster of two

bases. The architecture is centralized in the sense that the baseband pro-


cessing for a cluster of coordinated bases is performed at a single location

rather than in a distributed fashion across multiple bases. In the central-

ized architecture, each base is simply a remote radiohead which converts the

radio-frequency (RF) signal to baseband. The baseband signal from each base

(denoted as xb for base b = 1, 2) is sent over the backhaul to a centralized

baseband processor which jointly demodulates the signals. Estimates of the

users’ information streams, denoted by ub, are sent over another backhaul link

to the core network. We note that the centralized baseband processing could

be co-located with one of the radioheads, eliminating the need for backhaul

to send its baseband signal.

Downlink coordination can also be implemented using a centralized archi-

tecture as shown in the center subfigure of Figure 5.33. Here, the centralized

baseband processor generates the baseband signal sb that is transmitted over

base (radiohead) b. The information bit streams u1 and u2 are received from

the core network, and the local CSIT (or baseband signals for estimating the

CSIT) is obtained from each of the bases. The centralized processor there-

fore has global CSIT and can generate the baseband signal for the bases. For

coordinated precoding, the transmitted signal from each base is a function

of the data of its assigned user only. On the other hand, for network MIMO,

the transmitted signal from each base is a function of the data of all users

served by the coordination cluster.

Downlink coordination could be implemented using a distributed archi-

tecture, shown in the bottom subfigure of Figure 5.33, where each base per-

forms its own baseband signal processing. This architecture is similar to the

conventional architecture with independent bases, except that an enhanced

backhaul is required. For coordinated precoding, each base receives the in-

formation streams of its assigned users, as in the conventional architecture.

The enhanced backhaul is used for exchanging CSIT information between

bases. Under the assumption of slowly varying pedestrian channels, the total

backhaul requirement per base is only about 5% greater than the conven-

tional backhaul requirement because the bandwidth required for updating

the shared CSIT is minimal compared to the bandwidth of the data sig-

nals [160].

For network MIMO, the backhaul between the core network and the bases

requires higher bandwidth because the transmitted signal from each base

requires the data of both users. For coordinating a cluster of B bases, the

backhaul bandwidth between the core network and each base increases by

a factor of B to account for the user data. For small coordination clusters,

5.6 Summary 225

the distributed architecture backhaul bandwidth is lower for the distributed

architecture. For larger coordination clusters, the centralized architecture has

lower backhaul requirements [160].

Downlink network MIMO transmission requires signals from multiple bases

to arrive in a phase-aligned fashion at each user. Ideally, this requires tight

synchronization so there is no carrier frequency offset (CFO) between the lo-

cal oscillators at the base stations. Sufficient synchronization accuracy can be

achieved using commercial global positioning system (GPS) satellite signals if

the bases are located outdoors [161]. For indoor bases, the timing signal could

be sent from an outdoor GPS receiver or via CFO estimation and feedback

from mobiles [162].

5.6 Summary

In this chapter, we described the system model and simulation methodology

for evaluating the performance of MIMO cellular networks.

• The physical layer of a cellular network can be characterized in terms of

its system parameters (nature of the wireless channel, offered traffic pat-

terns, etc.), design constraints (bandwidth, power, cost, etc.), and design

outputs (base station and mobile requirements, air interface specification,

etc.). Within this framework, the system engineer designs the appropriate

antenna architectures, signaling strategies, and MIMO signal processing

techniques to make best use of the spectral and spatial resources.

• Three aspects of the system model were highlighted: sectorization, refer-

ence SNR, and cell wraparound. Sectorization is the radial partitioning of

a cell site, and its response as a function of azimuthal direction is mod-

eled as a parabolic function. The reference SNR is a single parameter that

conveniently encapsulates the effects of all parameters related to the link

budget. Cell wraparound is a technique that addresses the reduced inter-

ference at the edge of a finite cellular network in order to ensure uniform

performance statistics regardless of location.

• In conventional networks where base stations operate independently, in-

tercell interference can be mitigated in the spatial domain either implicitly

or explicitly using channel knowledge of the interferer. Coordinating the

transmission and reception of base stations potentially reduces interference

further at the expense of additional backhaul bandwidth.


Uplink centralized architecture

Downlink centralized architecture

Downlink distributed architecture

Baseband-RF

Basebandprocessing

Base 1

Corenetwork

Baseband-RF

Basebandprocessing

Base 2

Coordinated precodingNetwork MIMO

Baseband-RF

Radiohead 1

Baseband-RF

Radiohead 2Core

network

Centralizedbasebandprocessing

Baseband-RF

Radiohead 1

Baseband-RF

Radiohead 2

User 1

User 2

Corenetwork

Centralizedbasebandprocessing

Coordinated precodingNetwork MIMO

User 1

User 2

User 1

User 2

Fig. 5.33 Architectures for coordinated base stations. Uplink network MIMO can be im-

plemented with a centralized architecture. Downlink coordination techniques (coordinated

precoding and network MIMO) can be implemented with either a centralized or distributed

architecture.

• Proportional-fair and equal-rate criteria are two strategies for operating

a network to ensure fairness for all users. Uplink and downlink simula-

tion methodologies are described for each criteria. Interference mitigation

techniques using base station coordination, including coordinated precod-

ing and network MIMO, are described for the equal-rate criteria.

Chapter 6

Cellular Networks: Performance andDesign Strategies

In this chapter, we use the simulation methodology described in the previous

chapter to evaluate the network performance of different MIMO architectures.

We evaluate the performance of increasingly sophisticated systems, beginning

with independent bases each with a single antenna, moving to independent

bases with multiple antennas, and concluding with coordinated bases. In the

final section of this chapter, we synthesize the conclusions for the various

scenarios and outline a strategy for cellular network design under different

environments.

6.1 Simulation assumptions

The assumptions for the simulations are described here, following the frame-

work given in Section 5.1.

• Channel characteristics

We assume a pathloss coefficient γ = 3.76, which is consistent with the

Next-Generation Mobile Network (NGMN) Alliance’s simulation method-

ology for modeling a typical macrocellular environment [135]. For interference-

limited environments, the reference SNR is 30 dB. The shadow fading has

standard deviation 8 dB which is a typical assumption for macrocellular

networks [36] [135]. There is full shadowing correlation for co-located base

antennas that serve different sectors and zero shadowing correlation for the

users or bases otherwise. We assume the channel is frequency-nonselective

with zero delay spread. The channel is static over a coding block, so the

doppler spread is essentially zero. For most results, we assume rich scat-


227

228 6 Cellular Networks: Performance and Design Strategies

tering using an i.i.d. Rayleigh distribution. The exception is in Section

6.4.3 where the base antennas are totally correlated in order to implement

direction beamforming. Unless otherwise noted, the angle spread at the

base is zero to approximate the relatively small angle spreads found in

macrocellular networks.

• User statistics

Users are distributed uniformly throughout the network, and up to 30

users per site are considered. Mobile terminals are stationary or have

low-mobility to be consistent with the block fading assumption. The data

buffers for all users are always full.

• Fundamental constraints

The power constraint is modeled as part of the reference SNR parameter.

The carrier frequency affects the intercept of the pathloss model and is

therefore accounted for by the reference SNR parameter.

• Cost constraints

Because we are interested in establishing optimistic performance bounds,

we do not constrain the computational complexity transceivers. There-

fore we allow consideration of capacity-achieving DPC and MMSE-SIC

techniques for the downlink and uplink performance, respectively. Further

exploration of the relationship between performance and cost is beyond

the scope of this book.

• MAC and PHY-layer design

The air interface is assumed to be a packet-scheduled system, which is con-

sistent with next-generation cellular standards. We model the performance

of a single narrowband subcarrier of a wideband OFDMA air interface, and

we assume the air interface is designed so there is no intracell interference.

In other words, there is no interference between users served by a given

base that are scheduled on different subcarriers.

• Base station design

Bases are placed in a hexagonal grid of cells, and cell sites are partitioned

into S = 1, 3, 6, 12 sectors. Unless otherwise noted, the sector response

of the antennas is given by the model in (5.7). In scenarios with no base

station coordination, the networks are shown in Figure 6.9. In scenarios

with base station coordination, larger networks are used which are shown

in Figure 5.26. In either case, cell wraparound (Section 5.2.3) is used to

eliminate network boundary effects. Under the i.i.d. Rayleigh assumption,

the base station antennas need to have sufficient spacing (greater than

6.2 Isolated cell 229

10 wavelengths) or use dual-polarization. Under the correlated antenna

model, the spacing is assumed to be a half wavelength.

• Terminal design

The mobile antenna arrays are assumed to have an i.i.d. Rayleigh distribu-

tion. Because the mobiles are surrounded by local scatterers, this condition

can be achieved with half-wavelength spacing of the elements.

The simulation assumptions and parameters are summarized in Figure 6.1.

The proportional-fair criterion (with full-power downlink transmission and

power-controlled uplink transmission) is used for networks with independent

bases. The equal-rate criterion (with minimum sum power transmission) is

used for networks with coordinated bases. When needed, ideal CSIT is as-

sumed. This assumption is consistent with a TDD system with low-mobility

users. Likewise CSIR is known exactly, a reasonable assumption for indoor

applications [147] and for outdoor applications with slowly moving mobiles.

Distance-based pathloss

Shadow fading

Fast fading

Channel state information (CSI)

Traffic model

Intrasector interference

Pathloss coefficient:Intercept: Modeled by reference SNR

8 dB standard deviationFully correlated for co-located sectorsFully uncorrelated, otherwise

Time variation: Static, block fadingSpatial correlation: i.i.d. RayleighFrequency variation: Flat

Full buffer

Transmitter: Ideal, when applicableReceiver: ideal

None

Fig. 6.1 Parameters used for simulations in this chapter, unless otherwise noted.

6.2 Isolated cell

We first consider the SISO link performance in an isolated cell as a function

of only the distance-based pathloss using an omni-directional antenna. From


(5.5), the spectral efficiency of a user distance d from the base is

C = log2

[1 +

P

σ2

(d

dref

)−γ]

bps/Hz, (6.1)

where P/σ2 is the reference SNR at distance dref . Writing the noise variance

σ2 explicitly in terms of the noise power spectral density N0 and bandwidth

W , we can express the achievable rate as the bandwidth multiplied by the

spectral efficiency in (6.1):

R = W log2

[1 +

P

N0W

(d

dref

)−γ]

bps. (6.2)

Figure 6.2 shows the Shannon rate (6.2) as a function of the bandwidth W

for a reference SNR of P/σ2 = 0 dB given by the parameters for Downlink

B in Figure 5.10. For very low and very high values of SNR, the following

approximations are useful:

log2(1 + SNR ) ≈⎧⎨⎩SNR log2 e, if SNR � 1;

log2(SNR ), if SNR � 1.(6.3)

As the bandwidth decreases for a fixed power, the high-SNR approximation

in (6.3) is valid and yields

R ≈ W log2

[P

N0W

(d

dref

)−γ]. (6.4)

If the bandwidth W is low, then increasing it results in a nearly proportional

increase in the rate because the decreasing SNR inside the log term is in-

significant compared to the linear term W outside the log. This region is

known as the bandwidth-limited region.

If the bandwidth W is sufficiently high, the spectral efficiency becomes

a linear function of the SNR. In this power-limited region, increasing the

bandwidth does little to increase the rate, and eventually the rate reaches

an asymptotic limit. Using the low-SNR approximation in (6.3), the limiting

rate is

limW→∞

R =P

N0

(d

dref

)−γ

log2 e.

6.2 Isolated cell 231

100

102

104

106

108

1010

100

102

104

106

108

Bandwidth W (Hz)

Rat

e (b

ps)

d = 1000m

d = 10000m

Fig. 6.2 Rate versus bandwidth for a given transmit power (P/σ2 = 0 dB) and dis-

tance. As the bandwidth increases, the data rate saturates. As the distance increases, this

saturation occurs at a lower bandwidth.

101

102

103

104

105

100

102

104

106

108

1010

d (m)

Rat

e (b

ps)

0 dB=

20 dB=

10 Mbps, 50 km

Fig. 6.3 Rate versus distance for fixed transmit power (P/σ2 = 0 and 20 dB). For short

distances, SNR is high and rate is proportional to log SNR. At large distances, SNR is low

and rate is proportional to SNR. In order to achieve high data rates over a long distance,

the transmit power must be very high. To achieve 10 Mbps at 50 km in 10 MHz bandwidth,

the required transmit power is 400,000 W (P/σ2 = 67 dB).


For the parameters of Downlink B in Figure 5.10, P/σ2 = 0 dB and W =

10 MHz so that P/N0 = 107 Hz. The limiting rates are 14.4 Mbps and

2.5 kbps, respectively for d = 1000 m and 10,000 m.

In the range between 1 MHz and 100 MHz where current and future cellular

networks operate, the performance is somewhat power-limited at 1000 m and

severely power-limited at 10,000 m. Using the parameters for Downlink B,

7.2 watts with a 20 dB building penetration loss is not enough power for

additional bandwidth to increase the rate significantly.

Figure 6.3 shows the Shannon rate as a function of distance d for 10 MHz

and infinite bandwidth. The performance is bandwidth-limited for low d and

becomes power-limited as the receiver moves away from the transmitter. The

curves highlight the fact that at sufficiently large distances from the transmit-

ter, additional bandwidth will not increase the achievable rate significantly.

In this power-limited regime, very high power is required to achieve high

rates using SISO links. For example, to achieve 10 Mbps in 10 MHz at a dis-

tance of 50 km, a reference SNR of P/σ2 = 67 dB is required, corresponding

to 400,000 W transmit power. This magnitude underscores the difficulty of

achieving high data rates at large distances using SISO links without vio-

lating the laws of physics or Shannon theory. Even with MIMO techniques,

the size of the required power amplifiers is impractical, especially for uplink

transmission.

Conclusion: Even with abundant spectrum, serving a geographic area

with a single base is highly impractical due to the power requirements.

This fact provides the motivation for using a cellular network.

6.3 Cellular network with independent single-antenna

bases

In this section, we consider cellular networks where the bases have a single

antenna, and study the effects of reduced frequency reuse, reduced cell size,

and higher-order sectorization. As a baseline, we consider the performance

of a network with universal frequency reuse and a single sector per site. The

parameters for the four scenarios are summarized in Figure 6.4.

6.3 Cellular network with independent single-antenna bases 233

Sectors per site (S)


Transceiver

Users per sector (K)

Antennas per user (N)

Reference SNR (P/σ2)

Frequency reuse (F)

Transmission mode

Section

1

1

SISO

1

1

-30 to 30 dB

1

Full-power

6.3.1

1

1

SISO

1

1

10 dB

1

Full-power

6.3.2

1

1

SISO

1

1

-30 to30 dB

1, 1/3, 1/7

Full-power

6.3.3

3 6

1

SISO

1

1

30 dB

1

Full-power

6.3.4

12

Fig. 6.4 Parameters for downlink system simulations that assume independent, single-

antenna bases (Section 6.3).

6.3.1 Universal frequency reuse, single sector per site

We first study the downlink performance of a cellular network with universal

frequency reuse where a single omni-directional antenna is used at each site.

There is therefore one sector per site (S = 1). We assume that the channel

realization is dependent on only the distance-based pathloss and shadowing

and that there is no Rayleigh fading. A single user is assigned to each base

(K = 1), and as a special case of the proportional-fair methodology forK = 1,

we assume each base transmits with full power P . As a result, the geometry

(5.36) for a given user does not depend on the location of other users served

by other bases.

For very low transmit powers (low reference SNR), the normalized noise

power in the denominator of (5.36) dominates over the interference term:

(P/σ2)∑b =b∗

α2k,b � 1.

In this regime, the system is noise-limited, and increasing P/σ2 leads to

a significant increase in the geometry. For high reference SNRs where the


interference term dominates over the noise

(P/σ2)∑b=b∗

α2k,b � 1,

the system is interference-limited, and increasing P/σ2 results in a negligible

increase in the geometry.

For a given reference SNR P/σ2, the geometry is a random variable due to

the random placement of each user and the random shadowing. The cumula-

tive distribution function (CDF) of the geometry is shown in Figure 6.5 for

P/σ2 = −10, 0, 10 dB. In the range from P/σ2 = −10 dB to 0 dB, the system

is noise limited so that increasing the transmit power results in a significant

shift in the geometry CDFs. In the range from P/σ2 = 0 dB to 10 dB, the

system becomes interference limited so that the shift in the geometry curves

is much less.

For a given realization of the geometry Γ , the achievable rate is log2(1+Γ ).

Because there is one user assigned per site, the user rate is equivalent to the

base throughput. Figure 6.6 shows the resulting CDF of the throughput,

where the mean for each curve is indicated by the circle. The mean through-

put is plotted versus the reference SNR in Figure 6.7. For reference SNRs

greater than 10 dB, the performance is interference limited.

The reference SNR at which performance becomes interference limited

depends on the relationship between the interference and noise power. Here

we have assumed a single antenna per mobile. If the mobile had multiple

antennas, then spatial processing would reduce the interference power so that

the system would become interference limited at a higher reference SNR.

Conclusion: As the transmission power of all bases increases in a sys-

tem with independent bases, the intercell interference power begins to

dominate over the noise power, and the performance becomes inter-

ference limited. The maximum transmission power should be designed

so the performance is operating in the interference-limited regime be-

cause otherwise, spectral efficiency could be improved by increasing the

transmit powers.


-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

Geometry (dB)

= -10 dB= 0 dB= 10 dB

CD

F

Fig. 6.5 CDF of downlink geometry (5.36) under universal frequency reuse. Means are

indicated by the circles. As the reference SNR P/σ2 increases, the geometry becomes

interference limited. A 10-time increase in transmit power from P/σ2 = −10 dB to 0 dB

results in a significant shift in the CDF. From P/σ2 = 0 dB to 10 dB, the shift is much

less.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Throughput (bps/Hz)

CD

F

= -10 dB= 0 dB= 10 dB

Fig. 6.6 CDF of downlink throughput under universal frequency reuse. Throughput is

based on the geometry with no Rayleigh fading.


-30 -20 -10 0 10 20 300

0.5

1

1.5

2

2.5

3

Reference SNR (dB)

Mea

n th

roug

hput

(bps

/Hz)

Fig. 6.7 Mean throughput versus reference SNR under universal frequency reuse.

6.3.2 Throughput per area versus cell size

Suppose that a given reference SNR P/σ2 is achieved in a network with cells

of radius r1 (Figure 5.9) using transmit power P1. For a network with cells of

radius r2 < r1, the transmit power P2 required to achieved the same reference

SNR, assuming all other parameters are fixed, is obtained by solving

P1r−γ1 = P2r

−γ2 , (6.5)

where γ is the pathloss exponent. Hence for a fixed reference SNR, the re-

quired transmit power for a system with cell radius r2 is reduced by a factor

of (r1/r2)γ . (With γ = 3.76, a cell with radius r2 = 0.5r1 requires about a

factor of 14 less power.) If the number of users per cell is fixed and indepen-

dent of the cell radius, reducing the cell size increases the throughput per

unit area by a factor of (r1/r2)2 because the number of bases per unit area

scales with the cell radius squared.

Furthermore, we note that the scaling factor for the transmit power per

unit area is (r1/r2)γ−2. Hence for γ > 2, smaller cells increase the throughput


10-2 10-1 100 10110-2

10-1

100

101

102

103

104

intersite distance (km)

Thro

ughp

ut p

er a

rea

(bps

/Hz/

Km2 )

10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

Pow

er p

er a

rea

(wat

ts/K

m2 )

Fig. 6.8 The throughput per area (dashed line) and transmit power per area (solid line)

are plotted as a function of intersite distance for a cellular network with universal reuse

and one sector per cell. The reference SNR is fixed at P/σ2 = 30 dB. As the cell size

decreases, the throughput per area increases and the transmit power per area decreases.

Therefore smaller cells result in higher total throughput using lower total transmit power.

per area while requiring less total power per area. This observation provides

strong motivation for using small cells to serve densely populated areas with

high service demands. In practice, these small cells, sometimes known as fem-

tocells or picocells, should be deployed when the demand for service justifies

the total costs that include backhaul and baseband processing. These small

cells are especially effective when used indoors to provide indoor coverage

because the exterior building walls provide significant attenuation to reduce

interference between the indoor and outdoor networks.

Figure 6.8 shows the throughput per area and the total power per area as

a function of the intersite distance. At a reference distance of 1 km (corre-

sponding to an intersite distance of 2 km), a reference SNR of 10 dB can be

achieved using a transmit power of 0.72 watts, assuming 0 dB penetration

loss and with all other parameters given by the parameters for Downlink B in

Figure 5.10. From Figure 6.12, the mean throughput is 2.5 bps/Hz. The area


of a cell with radius 1 km is 6.9 km2. At an intersite distance of 2 km, the

throughput per area is 0.36 bps/Hz per km2, and the power per area is 0.10

watts per km2. Decreasing the intersite distance by a factor of 10 to 100 m,

the 10 dB reference SNR (and hence the same mean throughput per site) can

be achieved by transmitting with a factor 103.76 ≈ 5700 less power per site.

The density of sites increases by a factor of 100. Therefore the throughput

per area increases by a factor of 100 while the power per area decreases by a

factor of 57.

We emphasize that these scaling results assume that the number of users

per cell is fixed, independent of the cell size. If the number of users for the

total area of the network is fixed, then as the cell size decreases, the average

throughput would decrease significantly as some cells have no users to serve.

The scaling results assume that the pathloss coefficient is also independent

of the cell size. In practice, as the cell size decreases and the scattering chan-

nel becomes more like a line-of-sight channel, then the coefficient decreases.

For example, the pathloss coefficient for a line-of-sight microcell channel is

2.6 [36]. In this case, the additional interference would shift the geometry

distribution curve to the left, and the throughput per cell would decrease.

For the uplink, if the user density and pathloss coefficient are independent

of the cell size, then the uplink throughput per cell is also independent of the

cell size, and the same scaling results apply.

Conclusion: If the number of users per cell and the pathloss coeffi-

cient are independent of the cell radius, then as the radius decreases,

the throughput per area increases and the total transmitted power per

area decreases. In practice, the cost per site, including expenses for in-

frastructure, backhaul connections, and site leasing, need to be taken

into account when determining the cell site density.

6.3.3 Reduced frequency reuse performance

Sections 6.3.1 and 6.3.2 assumed universal frequency reuse where all B bases

transmit on the same frequency. Under reduced frequency reuse (Section

1.2.1), adjacent cells operate on different subbands, resulting in reduced in-

tercell interference. We study the performance of reduced frequency reuse


where the bandwidth W is partitioned into multiple subbands of equal band-

width. The parameter F ≤ 1 is known as the frequency reuse factor, and we

consider the cases of universal reuse (F = 1), reuse F = 1/3 with 3 subbands,

and reuse F = 1/7 with 7 subbands. To minimize co-channel interference, the

subbands are assigned using a reuse patterns shown in Figure 6.9.

For a given reuse factor F , we assume that each base transmits with power

P over bandwidth FW . If the noise spectral density is N0, then the noise

variance will be Fσ2 = FN0W . If we generalize the expression for the geom-

etry (5.36), the noise variance is reduced by a factor of F , and the geometry

of a user assigned to base b∗ is given by

(P/σ2)α2b∗

F +∑

b=b∗(P/σ2)α2

b

, (6.6)

where the summation occurs over bases operating on the same subband as

base b∗ and where we have simplified the notation by removing the user index.

f1f1

f1

F = 1

F = 1/3

F = 1/7

f2

f3

f1

f1

f3

f3

f1

f2

f2

f2f2

f3

f1f3

f1

f2

f2

f3

f1

f1

f3

f1

f6f7

f2

f5

f3

f4

f1

f6f7

f2

f5

f3

f4 f6f7

f2

f5

f3

f4

f1

f6f7

f2

f5

f3

f4f1

f6f7

f2

f5

f3

f4

f1

f6f7

f2

f5

f3

f4

f1

f6f7

f2

f5

f3

f4

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1

f1f1

f1

f1

f1

Fig. 6.9 Frequency assignment for cells under frequency reuse F . Under universal fre-

quency reuse (F = 1), all cells use the same frequency. Under reuse F = 1/3, the bandwidth

is partitioned into 3 bands f1, f2, f3. Under reuse F = 1/7, the bandwidth is partitioned

into 7 bands f1, . . . , f7.


-10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Geometry (dB)

CD

F

F = 1F = 1/3F = 1/7

Fig. 6.10 CDF of downlink geometry (6.6), parameterized by frequency reuse F , for

reference SNR P/σ2 = 30 dB. Means are indicated by the circles. Reducing frequency

reuse results in less interference and higher geometry.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate (bps/Hz)

CD

F

F = 1F = 1/3F = 1/7

(1,4)

(1,1)

Fig. 6.11 CDF of downlink rate/throughput (6.9), parameterized by frequency reuse F ,

for reference SNR P/σ2 = 30 dB. Means are indicated by the circles. Universal reuse

achieves the highest mean throughput. Under SISO, reduced reuse achieves higher cell-

edge rates, but with multiple receive antennas, universal reuse achieves highest cell-edge

rate.


-30 -20 -10 0 10 20 300

0.5

1

1.5

2

2.5

3

Reference SNR (dB)

Mea

n ra

te (b

ps/H

z)

F = 1F = 1/3F = 1/7

Fig. 6.12 Mean rate versus reference SNR for (1,1) links, parameterized by frequency

reuse F . The mean rate performance is interference-limited for P/σ2 ≥ 10 dB for all

frequency reuse factors F .

-30 -20 -10 0 10 20 300

0.05

0.1

0.15

0.2

0.25

0.3

Reference SNR (dB)

Cel

l-edg

e ra

te (b

ps/H

z)

F = 1F = 1/3F = 1/7

Fig. 6.13 Cell-edge rate versus reference SNR for (1,1) links, parameterized by frequency

reuse F . The cell-edge rate performance is interference-limited for F = 1 for P/σ2 ≥ 10 dB.

For reduced reuse, the cell-edge rate becomes interference limited at higher reference SNR.


Figure 6.10 shows the distribution of the geometry (6.6) for a reference

SNR of P/σ2 = 30 dB and for F = 1, 1/3, and 1/7. As F decreases, the

interference power decreases for a given user location, and as a result, the

geometry curves shift to the right.

If we let hb ∈ CN be the channel realization from base b to a user with

N antennas, the output of a maximal ratio combiner based on the desired

base’s channel hb∗ is

hHb∗x = |hb∗ |2sb∗ +

∑b=b∗

hHb∗hb + hH

b∗n, (6.7)

and the resulting SINR is

P/σ2||hb∗ ||2F +

∑b=b∗ P/σ

2 |hHb∗hb|2

||hb∗ ||2. (6.8)

Because each base uses only a fraction F of the total bandwidth, the spectral

efficiency is normalized by F and is given by

F log2

⎛⎝1 +

P/σ2||hb∗ ||2F +

∑b=b∗ P/σ

2 |hHb∗hb|2

||hb∗ ||2

⎞⎠ . (6.9)

Multiple antennas at the mobile (N > 1) provide combining gain, manifested

by the term ||hb∗ ||2 in the numerator, and interference-oblivious mitigation

(Section 5.3.1), manifested by the term |hHb∗hb|2 in the denominator.

Figure 6.11 shows the CDF of rates generated from (6.9) for reference

SNR P/σ2 = 30 dB using Rayleigh channels with N = 1 and 4. While

reduced reuse has an advantage over F = 1 for the SINR (6.8), this advantage

occurs inside the log term of (6.9), whereas the factor F outside the log term

results in a linear reduction of the rate. While universal reuse has the lowest

geometry, its rate performance is often better than the reduced reuse options

because of the relative advantage of the scaling term F = 1 outside the log.

In particular, the mean throughput is highest using universal reuse for both

N = 1 and N = 4.

For N = 1, universal reuse has the highest peak (90% outage) rate but

F = 1/7 reuse has the highest cell-edge (10% outage) rate. If multiple receive

antennas are used, the SINR (6.8) benefits from combining gain of the desired

signal and spatial mitigation of the interference. For N = 4, universal reuse

achieves the highest peak rate. For the cell-edge performance, because the

geometry of universal reuse (-2 dB) is in low-SNR regime of (6.3), a linear


increase in the SINR results in a linear increase in rate. On the other hand,

the cell-edge geometry of reuse F = 1/7 (13 dB) is in the high-SNR regime so

that a linear increase in SINR results in only a logarithmic improvement in

rate. Therefore the relative improvement in cell-edge rate due to combining

provides a more significant gain for F = 1, and its cell-edge rate is the best. In

an isolated link at low SNR, MRC combining results in a rate gain of about 4

(see Figure 2.7). However, in a system context, MRC provides additional gain

due to interference mitigation, so the total gain in cell-edge rate for F = 1 is

about a factor of 10.

For N = 1, the mean rate and cell-edge rate performance versus the ref-

erence SNR are shown respectively in Figures 6.12 and 6.13. The mean rate

performance is interference limited for all three values of F beyond 10 dB.

However, the cell-edge rate becomes interference limited at a higher reference

SNR when F < 1 because the noise power is still relatively large compared

to the interference power at the cell edge.

Note that if each cell in the network is assigned a unique frequency band,

the throughput performance of each cell would be noise limited. Therefore

an arbitrarily high throughput could be achieved by increasing the transmit

power. However, in a system with B bases, the bandwidth per base is W/B,

and the transmit power required to achieve a reasonable data rate would be

impractically high for any network having more than a few cells.

Conclusion: The mean throughput of a typical cellular network is

maximized under universal frequency reuse where each cell uses the

entire channel bandwidth. Reduced frequency reuse has the advantage

of better cell-edge rate performance if the mobiles have only a single

antenna, but this advantage is reduced with multiple-antenna mobiles.

6.3.4 Sectorization

Let us consider a downlink cellular network with S sectors per site andM = 1

antenna per sector. The sectors are assumed to be ideal for now and the

sector antenna response is given by (5.6). Universal reuse is assumed so that

the same frequency resources are reused in all sectors and cell sites. We

fix the total transmitted power per site so the power transmitted in each


sector is 1/S the total power. However, as a result of the sector antenna

gain, the received power from any base is independent of S. As a result, the

statistics of the channel variance α2b for the desired base sector is independent

of the number of sectors S. Likewise, the channel variance for any interfering

sector is independent of S. Therefore, because a user in a given location

receives non-zero power from exactly 1 sector per site as a result of the

ideal sector response, the resulting geometry distribution is independent of

S. Assuming that the channel statistics of each sector are identical, it follows

that for a given reference SNR P/σ2, the throughput distribution per sector

is independent of S. Therefore a network with S sectors per site with ideal

sectorization has a mean throughput S times that of a system with omni-

directional base antennas.

Assuming i.i.d. Rayleigh channels for the users, then as the number of

users increases asymptotically, the sum rate per sector scales with log logK

(Section 3.6). If we use a single panel antenna per sector, then with M sectors

per site, the sum rate per site scales with M log logK. In this sense, ideal

sectorization achieves the optimal sum-rate scaling of a cell site with M

transmit antennas. Compared to other scaling-optimal strategies such as DPC

or ZF beamforming, sectorization is much simpler to implement because it

transmits with only a single antenna in each sector. As a result, it requires

neither CSI at the transmitter nor computation of beamforming weights.

Similar throughput scaling can be achieved using non-ideal parabolic an-

tenna responses (5.7) if the intersector interference does not increase as S

increases. This condition can be met if we use the sector antenna parameters

in Figure 5.5. Figure 6.14 shows the resulting CDF of the geometry calcu-

lated using (5.36) for users distributed uniformly in a cellular network with

S = 1, 3, 6, 12 sectors under a reference SNR of P/σ2 = 30 dB. The geometry

distribution for S > 1 sectors per site is slightly degraded from the S = 1

geometry as a result of intersector interference. Because the shadowing real-

ization is identical for sectors associated with a given site, the peak geometry

occurs when a user is very close to a base and located in the boresight di-

rection of the serving sector. If the received power from the serving sector is

P ′, then, using the parameters in Figure 5.5, the power received from an in-

terfering sector co-located with the serving sector is P ′(3/S)100 for S = 3, 6, 12.

Therefore because there are S−1 interfering sectors per site, the peak geom-

etry is 100S3(S−1) . As seen in Figure 6.14, the peak geometry decreases a small

amount in going from S = 3 to 6 to 12 sectors. The peak geometry for S = 1

is much higher because there is no co-site intersector interference.


-10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Geometry (dB)

CD

F

S = 1S = 3S = 6S = 12

Fig. 6.14 CDF of geometry (5.36), parameterized by S, the number of co-channel sectors

per site. For S ≥ 3, the geometry is limited as a result of interference from other sectors

belonging to the same site. As a result of the sector response parameters in Figure 5.5, the

interference characteristics for S = 3, 6, 12 are similar.

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Throughput per site (bps/Hz)

CD

F

S = 1S = 3S = 6S = 12

Fig. 6.15 CDF of throughput per site based on the geometries from Figure 6.14. Mean

throughput per site approximately doubles in going from S = 3 to 6 sectors and again

from S = 6 to 12 sectors.


0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

18

20

3dB beamwidth (degrees)

Mea

n th

roug

hput

per

site

(bps

/Hz)

As

As = 20dB

= 26dB

S = 3

S = 6

S = 12

Fig. 6.16 Mean throughput per site versus sector beamwidth for S = 3, 6, 12 sectors per

site, parameterized by the sidelobe levels As. Zero-degree angle spread is assumed. For the

sidelobe levels given in Figure 5.5, the throughput performance is nearly optimal for the

corresponding beamwidths.

0 10 20 30 40 500

2

4

6

8

10

12

14

16

18

20


Mea

n th

roug

hput

per

site

(bps

/Hz) S = 3

S = 6S = 12

Fig. 6.17 Mean throughput per site versus RMS angle spread, using the sector parameters

in Figure 5.5. As the angle spread increases, the effective sector response becomes wider,

resulting in additional interference and degraded performance. For a given angle spread,

performance degradation for narrower sectors is more severe.


With regard to the overall geometry distribution, the CDF curve for S = 3

is similar in shape to that for S = 1 but shifted to the left as a result

of additional interference. For S = 3, 6 and 12, because the beamwidth

Θ3 dB and the sidelobe level As parameters are scaled by a half for each

doubling of S, the distribution of the geometry for a user k at a given angle

|θk,b∗ − ωb∗ | ≤ π/S with respect to its serving sector b∗ under S = 3 will be

similar to the distribution at the angle (θk,b∗ − ωb∗)/2 under S = 6 and at

the angle (θk,b∗ − ωb∗)/4 under S = 12. Therefore if the users are uniformly

distributed, the CDF of the geometry will be similar for S = 3, 6, 12.

Because the geometry distributions for S = 3, 6, 12 are similar, the result-

ing throughput distributions per sector are also similar. A throughput real-

ization per site is obtained by summing the throughput across S co-located

sectors for a given placement of users and channel realizations, and the re-

sulting CDF is shown in Figure 6.15. As a result of the central limit theorem

and the fact that the throughput realizations per sector are independent,

the throughput distribution per site becomes more Gaussian as S increases.

The mean throughput roughly doubles in going from S = 1 to 3. Because

the throughput per sector is largely independent of S (for S ≥ 3), the mean

throughput per site doubles in going from S = 3 to 6, and from S = 6 to

12. Therefore, as a result of the sector response parameters in Figure 5.5, the

intersector interference is independent of S, and the mean throughput un-

der higher-order sectorization with parabolic antenna patterns scales linearly

with S ≥ 3, as was observed in the ideal sectorization case.

What happens to the throughput if the sector beamwidth is wider or

narrower than the ones defined in Figure 5.5? Figure 6.16 shows the perfor-

mance mean throughput performance, still using parabolic sector responses

(5.7), but allowing the sector beamwidth (measured by the parameter Θ3 dB)

to vary from 5 to 75 degrees for S = 3, 6, 12. The sidelobe parameter As is

either 20 dB or 26 dB. This figure highlights the importance of designing the

sector response to match the number of sectors S. For S = 3 and As = 20 dB,

the throughput is essentially flat in the range of Θ3 dB from 50 to 70 degrees.

Therefore using a beamwidth of 70 degrees is reasonable. For S = 12 and

As = 26 dB, using a beamwidth of 18 degrees nearly maximizes the through-

put. The performance of higher-order sectorization, especially for S = 12, is

sensitive to the sidelobe levels and beamwidth.

The simulation results in Figures 6.15 and 6.16 assume that the chan-

nel has zero angle spread. For a non-zero angle spread, the effective sector

pattern can be computed as described in Section 5.2.1. As the angle spread


increases, the effective sector beamwidth increases (see Figure 5.8), resulting

in additional intercell and intersector interference. Figure 6.17 shows the im-

pact of higher angle spread on the throughput performance of sectorization.

The performance of higher-order sectorization is more sensitive because for a

fixed angle spread, there is more interference. In going from 0 to 10 degrees

angle spread, the throughput for S = 12 is reduced by almost 40% whereas

the throughput for S = 6 is reduced by only 20%. However, the absolute

throughput for S = 12 is still superior and is over twice the throughput of

conventional sectorization S = 3. As the angle spread increases beyond 40

degrees, there is very little throughput advantage in using S = 12 compared

to S = 6 sectors.

While sectorization is a relatively simple way of increasing throughput, a

few factors limit the sectorization order, preventing a designer from divid-

ing a cell into an arbitrarily large number of sectors. First, the azimuthal

beamwidth of the sector should be sufficiently large compared to the angle

spread of the channel so that the sidelobe energy falling into adjacent sectors

is tolerable. As a result, the geometry distribution will be independent of

S. Second, the physical size of the antenna should meet aesthetic and wind

load requirements. Third, if users are not uniformly distributed and there

are too few users in a sector, then higher-order sectorization would not be

justified. Fourth, the handoff rate may become too high if there are too many

sectors. Within these limits, higher-order sectorization is an effective way to

increase cell throughput and should be used as a baseline for comparison

against more sophisticated MIMO techniques. Recall that electronic beams

have several advantages over a panel implementation to address these factors,

including a smaller physical size and flexibility to steer beams and partition

power among them.

Conclusion: Sectorization achieves multiplexing gain for a cell site by

serving multiple users over different spatial beams. Ideal sectorization

(with no interbeam interference) achieves the optimal sum-rate scaling

per site. Using conventional, non-ideal sectorization with S = 3 as a

baseline, the throughput with higher-order sectorization scales in direct

proportion to S if the geometry distribution for a sector is independent

of S and if the channel angle spread is significantly less than the sec-

tor beamwidth. If the angle spread is too large, interbeam interference

diminishes the gains.

6.4 Cellular network with independent multiple-antenna bases 249

6.4 Cellular network with independent multiple-antenna

bases

Using multiple antennas, the bases can employ SU- and MU-MIMO tech-

niques to improve the system performance. Treating interference as noise, we

show that the area spectral efficiency exhibits the same linear scaling with

respect to the number of antennas as in isolated links. For a fixed number of

antennas per site, we study the tradeoff between implementing fewer sectors

per site with more antennas per sector and implementing more sectors per

site with fewer antennas per sector.

6.4.1 Throughput scaling

The capacity gain for MIMO channels compared to SISO channels was first

discussed in Chapter 1. The spatial multiplexing gains, which are defined for

asymptotically high SNRs, are observed for a wide range of SNRs. Figure 1.8

shows these gains can be achieved even for moderate SNRs, and Figure 1.9

shows that similar gains can be achieved in the low-SNR regime as a result

of combining. How are these gains reflected in a cellular system where the

performance is affected by interference and where the geometry of the users

is different?

Figure 6.19 shows the CDF of the throughput per site for S = 3 sectors

per site under ideal sectorization (5.6) and non-ideal sectorization (5.7) using

parameters from Figure 5.5. The performance is shown for the case of single-

user SISO transmission (M = 1, K = 1, N = 1), single-user closed-loop

MIMO transmission (M = 4, K = 1, N = 4), and sum-capacity multiuser

MISO transmission (M = 4, K = 4, N = 1). (A maximum sum-rate criterion

is used for MU-MISO.) The channels are assumed to be i.i.d. Rayleigh. These

parameters are summarized in Figure 6.18.

Under ideal sectorization, the distribution of the throughput per sector is

independent of the sectorization order S. Therefore the SU-SISO throughput

per site for S = 3 is a random variable equivalent to the summation of three

independent realizations of the SU-SISO throughput for S = 1 per site given

in Figure 6.15. As a result, the mean throughput for S = 3 is three times the

mean throughput for S = 1.


In going from single-user SISO transmission to closed-loop MIMO trans-

mission, the throughput increases as a result of combining gain for users

with lower geometry and spatial multiplexing for users with higher geometry.

At lower geometries (-10 dB to 0 dB), the MIMO system throughput gain

(where the interference is modeled as being spatially white, as in equation

(5.38)) can be inferred from Figure 2.7 for SNRs in the range of -10 dB to

0 dB. Due to the inequality in (5.43), the mean throughput obtained when

modeling the Rayleigh fading and colored interference is higher. Overall, the

gain at lower geometries is greater than 4 as a result of combining gain. For

example, at the cell edge (10% outage) throughput gain is about a factor of

6.8 in going from 2.5 bps/Hz to 17 bps/Hz. For users with higher geometries,

the MIMO throughput gain is about a factor of 4 due to spatial multiplexing.

(The peak throughput at 90% outage increases from 14 bps/Hz t 56 bps/Hz

for a gain of 4.) Overall, the mean throughput increases by about a factor

of 4.5, from 7.8 bps/Hz to 35 bps/Hz. (If we modeled the interference as

spatially white (5.38), then the mean throughput increases by a factor of 4.3,

from 7.2 bps/Hz to 31 bps/Hz. This result is not shown in Figure 6.19.)

Under MU-MISO transmission with ideal sectorization, multiple users are

distributed across the cell and geometry realizations drawn from the CDF

in Figure 6.14 for S = 1. As a result of multiuser diversity, the through-

put distribution would have less variance than the case where the multiple

users have the same realization. By coordinating the reception of users lo-

cated in the same location, the throughput would be improved, and this case

would correspond to the SU-MIMO curve in Figure 6.19. Therefore in going

from SU-MIMO transmission to MU-MISO transmission, the CDF becomes

steeper and the mean throughput decreases.

Compared to the performance with ideal sectorization, the performance

using the non-ideal sector response is degraded as a result of intersector

interference. Because the degradation in the geometry is more significant at

the upper tails as shown in Figure 6.14, the degradation in throughput is

likewise more significant at the upper tails of its CDF.

The mean throughput for the case of nonideal parabolic sector responses

(5.7) is plotted in Figure 6.20 as a function of M for M = 1, 2, 4. (For M = 2,

SU-MIMO transmission assumes K = 1 and N = 2, and MU-MIMO trans-

mission assumesK = 2 and N = 1.) In addition to the performance under the

maximum sum-rate criterion for the downlink, corresponding site through-

put performance is also shown for the uplink and for both the downlink and

uplink under the equal-rate criterion. The linear scaling with respect to M is


apparent from this figure. Under the equal-rate criteria, the throughput per

site is lower because the bases transmit with less than full power. Despite

the differences in the absolute multiuser throughput between the two crite-

ria, the scaling gains are very similar. Under the proportional-fair criterion,

the multiuser downlink performance is reduced compared to the maximum

sum-rate criterion. For M = 4, K = 4, N = 1, the mean throughput per site

is 19 bps/Hz, and the throughput gain with respect to SU-SISO is 3.8. These

results are shown later in Figure 6.26 but not in Figure 6.20.

Conclusion: When MIMO techniques are implemented in a cellular

network, the average throughput increases linearly with respect to the

number of antennas. These gains are achieved through spatial multi-

plexing for users with high geometry and through combining for users

with low geometry.

6.4.2 Fixed number of antennas per site

In this section, we fix the number of base antennas per site and compare the

performance for different numbers of sectors per site. We first study single-



Transceiver




Frequency reuse (F)

Transmission mode

3

1, 2, 4

SU-MIMO

1

N = M

Full-power

3

1, 2, 4

MU-MIMO

K = M

1

Full-power,Min. sum power

30 dB

1

Fig. 6.18 Parameters for system simulations that show throughput scaling as a function

of the number of base antennas per sector M (Section 6.4.1).


0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CD

F

Throughput per site (bps/Hz)

M=1, SU

M=4, SU

M=4, MU

IdealParabolic

Fig. 6.19 CDF of throughput per site under different sectorization and interference as-

sumptions. Under ideal sectorization, performance improves in going from white to colored

interference as a result of Jensen’s inequality. Under colored interference, performance de-

grades in going from ideal to nonideal parabolic sector responses as a result of intersector

interference.

1 2 40

5

10

15

20

25

Thro

ughp

ut p

er s

ite (b

ps/H

z)

SU ULSU DLMU ULMU DL

UL: 4.2xDL: 4.1x

UL: 4.2xDL: 4.2x

UL: 4.4xDL: 4.5x

UL: 2.1xDL: 1.9x

UL: 2.1xDL: 2.1x

UL: 2.2xDL: 2.2x

Maximum sum-rate criterion

Equal-rate criterion

Fig. 6.20 Average throughput (under the proportional-fair criterion) and 10% outage

throughput (under the equal-rate criterion) versus M , the number of antennas per sector.

Simulation parameters are given by Figure 6.18, under the assumption of parabolic sector

responses and colored interference. For both criteria and for both uplink and downlink,

throughput scales with M .


user MIMO techniques for six antennas per site. We consider S = 3 sectors

per site with M = 2 antennas per sector and S = 6 sectors per site with

M = 1 antenna per sector. We then study multiuser MIMO techniques for

twelve antennas per site with S = 3, 6, 12 sectors per site and, respectively,

M = 4, 2, 1 antennas per sector.

6.4.2.1 Single-user MIMO, six antennas per site

Fixing the number of antennas and users per site to be six, we consider con-

ventional sectorization (S = 3 sectors per site, M = 2 antennas per sector,

K = 2 users per sector) and higher-order sectorization (S = 6 sectors per

site, M = 1 antenna per sector, K = 1 user per sector). The options and pa-

rameters are summarized in Figure 6.21. Sector responses are given by (5.7)

and parameters in Figure 5.5. Multiple antennas at a given base (sector) are

assumed to have the same boresight direction and to be spatially uncorre-

lated. Therefore they could be implemented using a cross-polarized (Div-1X)

configuration or two widely separated columns (Tx-Div) as discussed in Sec-

tion 5.5. For the S = 3 sector case, we evaluate the performance of Alamouti

transmit diversity, open-loop spatial multiplexing, and closed-loop spatial

multiplexing. We also consider SIMO transmission (M = 1) as a baseline

for comparison. Mobile users are assumed to have N = 2 antennas, and the

channels are spatially i.i.d. Rayleigh.

Single user per sector

We first consider the distribution of the user rates when there is single user per

sector. The distribution is generated over random user locations, shadowing

realizations, and Rayleigh channel realizations. The CDF for the five cases is

shown in Figure 6.22. Compared to the S = 3,M = 1 baseline, the transmit

diversity distribution has a smaller variation so there is a slightly improved

cell-edge performance and lower peak-rate performance. The mean rate is

slightly lower and is a consequence of the Rayleigh channel realizations of

the interfering cells. (If we model the interference using the average power

based on αk,b instead, the mean throughput using transmit diversity would

be slightly higher than the baseline’s.) While transmit diversity improves the

link reliability, the reduction in the range of channel variations is actually

detrimental to scheduled systems that rely on large channel variations [157].

Furthermore, in contemporary scheduled cellular networks operating in wide


bandwidth, transmit diversity has limited value because there are other forms

of diversity including multiuser diversity, receiver combining diversity and

frequency diversity [163].

For CL-MIMO, multiplexing gain is achieved at high geometry and there

is a significant improvement in the peak rate compared to the baseline. At

low geometries, CSIT allows the transmitter to shift power to the most favor-

able eigenmode so that CL-MIMO achieves a significant cell-edge rate gain as

well. For OL-MIMO, multiplexing gain is achieved at high geometry, and its

peak rate nearly matches that of CL-MIMO because their performances are

equivalent for asymptotically high geometry. At low geometries, OL-MIMO

transmits a single stream using diversity and its performance is equivalent

to (2,2) transmit diversity. In practice, because CL-MIMO requires CSIT,

the performance advantage over OL-MIMO is reduced if the CSIT becomes

less reliable, for example if the mobile speed increases. Using higher order

sectorization with S = 6 sectors, the geometry distribution is similar to that

of the baseline (Figure 6.14), so the rate distribution is similar but with a

slightly degraded performance.



Transceiver




Frequency reuse (F)

Transmission mode

3

1

SIMO

2

3

2

Tx DiversityOL-MIMOCL-MIMO

2

2

30 dB

1

Full-power

6

1

SIMO

1

Fig. 6.21 Parameters for system downlink simulations that assume up to 6 antennas per

site (Section 6.4.2.1).


0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

User rate (bps/Hz)

CD

F

S = 3, M = 1, SIMOS = 6, M = 1, SIMOS = 3, M = 2, TxDivS = 3, M = 2, OL-MIMOS = 3, M = 2, CL-MIMO

Fig. 6.22 CDF of user rate, 1 user per sector (N = 2 antennas per user), for up to 6

antennas per site. CL-MIMO achieves the best performance because it has M = 2 antennas

with ideal CSIT.

0

5

10

15

20

25

30

35

17

8.8

2.2

16

8.3

2.5

22

10

2.7

23

12

3.8

32

16

4.0

bps/

Hz

SIMOS = 3M = 1

TxDivS = 3M = 2

OL-MIMOS = 3M = 2

CL-MIMOS = 3M = 2

SIMOS = 6M = 1

6 antennas per site

Normalized peak user rate

Normalized mean user rate, Mean throughput per site

Normalized cell-edge user rate

Fig. 6.23 Mean throughput and normalized user rate performance for 6 users per site and

up to 6 antennas per site. While the performance per sector of higher-order sectorization

is inferior to the other options, its performance per site is the best.


M users per sector

We emphasize that the rate distributions in Figure 6.22 are from the perspec-

tive of the mobile user, and the distributions assume there is a single user

per sector. To make fair performance comparisons for different architectures

that have different values of S, we consider the performance for a fixed total

number of users per site. We set the number of users per site to be six and

assume an even distribution of users among the sectors. In general, there

are M users per sector. For a system with S = 3 sectors per site, there are

K = 2 users per sector, and for system with S = 6 sectors per site, there

are K = 1 users per sector. For the case of S = 3, the two users in a sector

are served fairly using a round-robin scheduler because their channels are

assumed to be static (Section 5.4.1). (If the channels were time varying, then

a channel-aware scheduling algorithm could provide multiuser diversity gains

over round-robin scheduling.)

For a single user per sector, if the inverse CDF of the user rateR′ for a givenprobability α ∈ [0, 1] is F−1

R′ (α) = x, then for two users served in a round-

robin fashion, the inverse CDF of the user rate R would be F−1R (α) = x/2.

Therefore for K = 2 users per sector, the CDF of user rate for each of

the architectures with S = 3 could be obtained by shifting the appropriate

curve in Figure 6.22 to the left so that for a given value on the y-axis, the

value on the x-axis is reduced by a factor of 1/2. For example, for SIMO

(S = 3, M = 1) with K = 2 users per sector, the cell-edge user rate would be

F−1R (0.1) = 0.4 bps/Hz, the peak user rate would be F−1

R (0.9) = 2.8 bps/Hz,

and the mean user rate would be E(R) = 1.5 bps/Hz.

As defined in Section 5.4.3.1, the normalized peak user rate (SK ×F−1R (0.9)) and normalized cell-edge user rate (SK×F−1

R (0.1)) allow for mean-

ingful comparisons with the mean throughput per site (SK × E(R)). These

statistics are plotted in Figure 6.23. The relative performances of the tech-

niques with S = 3 are the same in this figure as they were in 6.22 because

the number of users per sector is the same (K = 2). However, while the

SIMO performance of S = 6 is similar to that of S = 3 in Figure 6.22, the

system-level performance of higher-order sectorization (S = 6) is superior

with almost a doubling in the values of each metric. The performance of

higher-order sectorization is also superior to the CL-MIMO performance for

all three metrics. Not only does higher-order sectorization achieve the best

performance, its signal processing complexity is less than any of the MIMO

techniques because it uses single-antenna transmission and MRC combining.


While single-user spatial multiplexing is known to provide higher peak user

rates, it may be surprising that S = 6 SIMO achieves the highest normalized

peak user rate. The reason is that while the peak rate achieved under S = 6

SIMO will be lower for each transmission interval compared to S = 3 CL-

MIMO, its normalized peak rate will be higher because, for a fixed number

of users per site, each user under S = 6 will be served twice as often.

Another way to understand the gains of sectorization is to note that by

doubling the number of sectors per site, the spectral resources are doubled

and the performance metrics improve by a factor of two without having to

increase the number of antennas per mobile. As shown in Figure 6.24, the

mean throughput for SU-SIMO with S = 6,M = 1,K = 1, N = 2 (16

bps/Hz) is about twice that of SU-SIMO with S = 3,M = 1,K = 2, N = 2

(8.8 bps/Hz). On the other hand, the SU-MIMO system with S = 3,M =

2,K = 2, N = 2 uses the same resources as the S = 6 SIMO system, but its

throughput (12 bps/Hz) is about twice that of the single-user SISO system

with S = 3,M = 1,K = 2, N = 1 (5.3 bps/Hz, from 6.19).

Conclusion: Transmit diversity provides minimal gains in cell-edge

rate over single-antenna transmission and is of limited value in contem-

porary cellular networks which already have other forms of diversity.

Closed-loop SU-MIMO is most advantageous with respect to open-loop

SU-MIMO at low geometries and at low doppler rates (e.g., pedes-

trian environments). At high geometries, the performance gap vanishes.

Closed-loop (2,2) MIMO provides modest improvements (30%) in peak

rate and throughput over (1,2) SIMO. However, for the same number of

antennas per site, higher-order sectorization provides even better peak-

rate and throughput performance. For systems with many uniformly

distributed users, doubling the number of sectors per site doubles the

throughput and user metrics.

6.4.2.2 Multiuser MIMO, twelve antennas per site

We now consider 12 antennas per site and the option of more advanced mul-

tiuser MIMO techniques. Will higher-order sectorization be competitive with

these more advanced MIMO techniques? The transceiver options are sum-

marized in Figure 6.25. Each cell site is split into S = 3, 6, or 12 sectors


S = 3 M = 1

5.3 bps/Hz

K = 2N = 1

S = 3 M = 1

8.8 bps/Hz

K = 2N = 2

S = 6 M = 1

16 bps/Hz

K = 1N = 2

S = 3 M = 2

12 bps/Hz

K = 2N = 2

Gain from combining

Gain from combining and spatial multiplexing,

factor of ~2

Gain from sectorization,factor of ~2

SISO SIMO Higher-order sectorization

MIMO

Fig. 6.24 The throughput of higher-order sectorization is approximately double that of

the SIMO system, while the throughput of the MIMO system is approximately double that

of the SISO system. Therefore for 6 antennas per site, higher-order sectorization results in

a higher throughput than the MIMO system.

using, respectively, M = 4, 2, and 1 antennas per sector, where multiple

antennas for a given sector are assumed to be uncorrelated. Therefore the

M = 2 and M = 4-antenna systems could be implemented using, respec-

tively, a Div-1X and Div-2X antenna configurations (Section 5.5). Both the

uplink and downlink performance are studied, and for both cases, the respec-

tive capacity-achieving strategy assuming ideal CSIT is used: DPC for the

downlink and MMSE-SIC for the uplink.

For S = 12, single-antenna reception and transmission are used respec-

tively for the downlink and uplink. As before, the sector responses are given

by (5.7) and parameters in Figure 5.5. Twelve users, each equipped with

N = 1 or 4 antennas, are assigned to each site. For each sector of the

S = 3, 6, 12-sector configurations, there are respectively K = 4, 2, 1 users


distributed uniformly in the sector. CL SU-MIMO is used as the baseline for

comparison.



Transceiver




Frequency reuse (F)

Transmission mode

3

4

SU-MIMOMU-MIMO

4

6

2

SU-MIMOMU-MIMO

2

1 or 4

30 dB

1

Full-power

12

1

SU-SIMO (DL)SU-MISO (UL)

1

Fig. 6.25 Parameters for uplink and downlink system simulations that assume 12 anten-

nas per site, with S = 3, 6 or 12 sectors per site (Section 6.4.2.2). Closed-loop MIMO is

used for single-user transmission. Capacity-achieving transceivers are used for MU-MIMO.

Single-user SIMO and MISO are used for downlink and uplink transmission, respectively.

Figures 6.26 and 6.27 show the normalized user rate statistics and mean

throughput per site for the various transceiver options with N = 1 and

N = 4, respectively. The uplink and downlink results for a given architecture

are similar because the underlying geometry distributions are similar when

power control is used for the uplink (Figure 5.24). However, as a result of

the specific power control parameters used, the uplink has lower cell-edge

rates than the downlink. Because of the similarity between the uplink and

downlink performance, we focus our discussion on the downlink results.

Single-antenna mobiles, N = 1

For now, we consider the downlink results in Figure 6.26 where N = 1. We

first compare the performance of SU-MISO and MU-MISO for S = 3,M = 4.

The MU performance is superior to the SU performance because the latter

is a special case of the former where transmission is always restricted to a

single user at a time. For SU-MISO, the K = 4 users per sector are served in


0

10

20

30

40

50

60

19

10

3.7

39

19

5.9

30

15

3.4

40

19

4.1

46

19

1.9

23

13

4.8

55

23

3.3

35

17

2.6

52

21

2.0

51

20

1.0

bps/

Hz

SU-SISO

SU-MISO

MU-MISO

SU-MISO

MU-MISO

S = 3M = 4

S = 6M = 2

S = 12M = 1

Downlink

SU-SISO

SU-SIMO

MU-SIMO

SU-SIMO

MU-SIMO

S = 3M = 4

S = 6M = 2

S = 12M = 1

Uplink




N = 1

Fig. 6.26 Mean throughput and normalized user rate performance for 12 antennas per

site, N = 1. For both uplink and downlink, higher-order sectorization with S = 12 achieves

throughput performance comparable to MU-MIMO with significantly lower signal process-

ing complexity.

0

10

20

30

40

50

60

70

80

90

46

24

9.4

72

36

14

61

33

13

73

40

16

76

46

19

48

24

7.4

85

37

6.4

62

31

7.6

79

37

7.3

76

39

8.4

bps/

Hz

SU-SIMO

SU-MIMO

MU-MIMO

SU-MIMO

MU-MIMO

S = 3M = 4

S = 6M = 2

S = 12M = 1

Downlink

SU-SIMO

SU-MIMO

MU-MIMO

SU-MIMO

MU-MIMO

S = 3M = 4

S = 6M = 2

S = 12M = 1

Uplink




N = 4

Fig. 6.27 Mean throughput and normalized user rate performance for 12 antennas per

site, N = 4. Higher-order sectorization performs well, and multiple user antennas increase

the cell-edge rate significantly.


a round-robin fashion because the channel is assumed to be static (Section

5.4.1). A single user is served during each transmission interval, but because

CSIT is used, the M = 4 antennas provide combining gain. For MU-MISO

transmission, multiplexing gain can be achieved by serving multiple users

simultaneously. Insights for the relative performance of SU- and MU-MISO

can be gained from the performance results in Figure 3.15 for N = 1. As

the SNR increases, the gain of the BC sum-capacity versus the TDMA sum-

capacity increases, and this is reflected in Figure 6.26 where the peak rate

gains for MU-MISO compared to SU-MISO (39 bps/Hz versus 19 bps/Hz) are

higher than the cell-edge rate gains (5.9 bps/Hz versus 3.7 bps/Hz). Overall,

the MU transmission results in about a factor of two improvement in mean

throughput versus SU transmission (19 bps/Hz versus 10 bps/Hz).

For S = 6,M = 2, the MU-MISO performance is superior to the SU-MISO

performance because SU transmission is a special case of MU transmission.

However, the relative performance advantage is reduced when compared to

the S = 3 sector case because the multiplexing gain for MU-MISO is reduced

from 4 to min(M,KN) = min(2, 4) = 2.

Considering the performance of higher-order sectorization with S = 12

and M = 1, we see that its mean throughput (19 bps/Hz) is the same as

the throughput for MU-MISO S = 3 and MU-MIMO S = 6. Higher-order

sectorization achieves this performance using a much simpler single-antenna

transmission strategy compared to the dirty paper coding strategy required

in general for the multiuser transmission. Some insights into the relative

performance of MU-MISO S = 3 and SU-SISO S = 12 can be gained from

Figure 3.14 by considering the top BC curve for N = 1. This curve shows that

the relative performance of the ((4,1),4) BC sum-capacity compared to the

(1,4) SU capacity is higher in the low-SNR regime because of the advantage

of transmitter combining for the BC. This trend is reflected in the superior

cell-edge performance for MU-MISO with S = 3 compared to SU-SISO with

S = 12 (5.9 bps/Hz versus 1.9 bps/Hz). We note that the relative gains in

Figure 3.14 should be interpreted as the throughput per sector in the system

context. The equivalent mean throughput per site for S = 12 and MU-MISO

S = 3 implies that the mean throughput per sector should be a factor of 4

greater for MU-MISO, and this is consistent with the ratio of average sum

rates in Figure 3.14 being between 5 and 3.7 for the relevant range of SNRs.

The concluding observation we make in Figure 6.26 is that the mean

throughput and peak rate performance with higher-order sectorization S =

12 is comparable to the performance of MU-MISO with fewer sectors per site.


However, the cell-edge performance of higher-order sectorization is inferior.

Multi-antenna mobiles, N = 4

If the transmitter has multiple antennas, then multiple antennas at the mobile

allow the possibility of spatial multiplexing at high geometry. For S = 3, the

spatial multiplexing gain for SU-MIMO (min(M,N) = 4) and MU-MIMO

(min(M,KN) = 4) are the same, and the gain in peak rate due to multiuser

transmission (46 bps/Hz versus 72 bps/Hz) is reduced compared to the MISO

case (19 bps/Hz versus 39 bps/Hz). A similar trend can be observed for

between SU-MIMO and MU-MIMO for S = 6.

For single-antenna transmitters in the case of higher-order sectorization

(S = 12), multiple antennas at the mobiles can be used to achieve combin-

ing gain with respect to the signal from the desired base and interference-

oblivious mitigation with respect to interference from other bases. For higher-

order sectorization, using N = 4 antennas improves the cell-edge user rate by

a factor of 10 (19 bps/Hz versus 1.9 bps/Hz). (The same gain was achieved

for the cell-edge rate in Figure 6.11 for F = 1.)

If the transmitter has multiple antennas, the marginal gains achieved from

multiple receive antennas are less. For example, the cell-edge user rate for

S = 3,M = 4 increases by about a factor of 2.5 (from 3.7 to 9.4 bps/Hz).

Overall, higher-order sectorization achieves the best performance for all

three metrics, and the implementation requirements are significantly less than

those for MU-MIMO. Higher-order sectorization does not require CSIT and

uses simple single-antenna transmission. In contrast, CSIT is required for

DPC when S = 3 or 6, and the computational complexity of DPC is very

high. Furthermore, DPC performance depends on the reliability of the CSIT,

and it is much less robust than single-antenna transmission. (For the uplink

for S = 12, a single antenna is used to detect a single user, and the receiver

complexity is lower than for S = 3 or 6 where an MMSE-SIC receiver is

used.)

Conclusion: If the number of antenna elements per site is fixed at

12 and if the channel angle spread is sufficiently narrow, higher-order

sectorization is an effective, low-complexity technique to achieve high

average throughput, even when compared to far more sophisticated MU-

MIMO techniques. MU-MIMO provides performance gains over SU-

MIMO by exploiting multiplexing of streams across multiple users. Mul-


tiple antennas at the terminal provide higher gains for users with lower

geometry and for bases with fewer antennas. For a given transceiver ar-

chitecture, the performance between the uplink and downlink is similar

because the underlying geometry distributions are similar.

6.4.3 Directional beamforming

The superior performance of higher-order sectorization S = 12 described

above in Section 6.4.2.2 assumes that the sector response has very good

sidelobe characteristics that result in minimal intersector interference. A

parabolic sector response (5.7) with parameters in Figure 5.5 can be achieved

using a sector antenna implemented using a ULA-4V array (Section 5.5.2).

Twelve of these sector antennas are deployed per site, and the result is phys-

ically unwieldy and visually unattractive, as shown in Column C of Figure

5.28.

As an alternative, we can create the sector beams electronically using a

more compact array, as shown in column G of Figure 5.28, by employing

three uniform linear arrays per site. Each array consists of M = 4 or 8

antennas with half-wavelength spacing, with each element having the same

boresight direction (Figure 5.4, S = 3), the same element response (5.7) and

the same parameters (S = 3 in Figure 5.5. We assume a line-of-sight channel

model where base antennas are fully correlated and the relative phase offsets

between antennas are a function of the signal direction (2.79). Each array

forms four fixed beams with directions shown in Figure 6.28.

Using a Dolph-Chebyshev design criterion (Section 4.2.3), the beamform-

ing weights can be designed to point the top array in Figure 6.28 towards 105.

A separate set of weights can be designed for 135 degrees. The beam weights

for 45 and 75 degrees can be derived through symmetry, and the weights for

these four beams can be used by the other two M -antenna arrays to generate

their beams. The resulting beam responses for 105 and 135 degrees are shown

in, respectively, the top and bottom subfigures of Figure 6.29. In the top sub-

figure, the sidelobe characteristics for M = 4 and 8 elements are similar, but

the beamwidth is narrower with more elements. With M = 8, the beamwidth

is in fact slightly narrower than the parabolic response beamwidth (5.7) for

S = 12. The bottom subfigure is for the beams steered towards 135 degrees.


45o

…

……

75o105o

135o

165o

-165o

-135o

-105o

15o

-15o

-45o

-75o

M antennas

Fig. 6.28 Twelve fixed beams are created per site under directional beamforming. Four

beams are formed using a linear array of M = 4 or 8 antennas. Directions are measured in

degrees with respect to the positive x-axis.

The sidelobe characteristics for M = 4 are worse because the desired beam

is 45 degrees off the broadside direction of the elements. The beam formed

with M = 8 elements has better sidelobes and narrower beamwidth.

The twelve beams formed with the ULAs replace the panel sector responses

for S = 12, and the mean throughput per site versus the channel angle spread

is shown in Figure 6.31 for the case of N = 2 antennas per user. Compared to

the performance of theM = 4-element ULA system, the S = 12-sector system

has about a 25% gain in throughput for zero-degree angle spread as a result

of narrower beamwidth and lower sidelobes. However, the ULA performance

is similar to that of the S = 3,M = 4 system using DPC. Therefore using

the same number of antennas per sector, the throughput performance for

fully uncorrelated and fully correlated base antennas is similar. Using M =

8 elements in the ULA, even though the beams are not the same as the

S = 12 response, the resulting SINR distributions are similar, and the mean

throughput performances are nearly identical for the range of angle spreads

considered.

As an alternative to the ULA architecture, the architecture proposed

in [164] uses 24 V-pol columns arranged in a circle to form twelve beams for

sectorization. As shown in Figure 6.32, each beam is generated using seven

adjacent columns, where broadside direction of the center column is pointing

in the desired direction of the sector. Due to the symmetry of the circular


0 50 100 150 200 250 300 350-50

-40

-30

-20

-10

0

Direction of user, θ (degrees)

Bea

m re

spon

se, G

(θ) (

dB)

…

105o

0 50 100 150 200 250 300 350-50

-40

-30

-20

-10

0

Direction of user, θ (degrees)

Bea

m re

spon

se, G

(θ) (

dB)

ParabolicDolph-Chebyshev, M=4Dolph-Chebyshev, M=8

Parabolic Dolph-Chebyshev, M=4Dolph-Chebyshev, M=8

…

135o

Fig. 6.29 Response of beams formed using uniform linear arrays withM = 4 or 8 antennas

per sector (with S = 3 sectors per site). Four fixed beams are formed per sector in order

to approximate the a S = 12-sector site using panel antennas to create parabolic sector

response. The response of the ULA beam with M = 4 in the top subfigure is the same as

the Dolph-Chebyshev response in Figure 4.10.

architecture, the response of each beam is identical with respect to the corre-

sponding sector direction, unlike in the case of multiple beams formed with a

ULA. It was shown that the measured response of a circular array prototype

achieves a similar response as S = 12 in Figure 5.5 [164]. Therefore the per-

formance shown in Figures 6.26 and 6.27 for S = 12 could be achieved using

a much smaller circular array architecture compared to 12 ULA-4V panels.




Transceiver

Users per sector



Frequency reuse (F)

Transmission mode

3 (12 virtualsectors)

4, 8

SIMO

1 per virtualsector

2

30 dB

1

Full-power

12

1

SIMO

1

…

…

…

Fig. 6.30 Parameters for system simulations that assume S = 12 sectors per site, imple-

mented with either multibeam antenna array or sector antennas (Section 6.4.3). At each

site, each of the three multibeam antenna arrays creates four virtual sectors.

0 10 20 30 40 500

5

10

15

20

25

30


Mea

n th

roug

hput

per

site

(bps

/Hz)

S = 12, M = 1S = 3, M = 8S = 3, M = 4

Fig. 6.31 Mean throughput per site versus angle spread. M = 4 or 8 antennas are used

per sector (S = 3) to form beams (see Figure 6.29. Each user has N = 2 antennas. The

throughput performance with M = 8 antennas is identical to the performance with the

parabolic S = 12 pattern. The throughput performance with M = 4 is comparable to the

MET (S = 3, M = 4) performance in Figure 6.26 for zero angle spread.


Sector 1

Sector 2

Fig. 6.32 A circular array architecture can be used to create beams for S = 12 virtual

sectors. The array consists of 24 V-pol columns, and each beam is generated using seven

adjacent columns, where broadside direction of the center column is pointing in the desired

direction of the sector.

Conclusion: In low angle spread environments, an array of closely

spaced base station antennas can be used to create directional beams

for sectorization. With enough antennas in the array, the resulting per-

formance is comparable to a system that uses a single antenna for each

sector, and because a single array can form multiple beams, the array

architecture is more compact. Also, the beams are more flexible and

can adapt to non-uniform loading.

6.4.4 Increasing range with MIMO

While we have focused on using MIMO as a means of increasing the spectral

efficiency per site for a fixed cell size, it is also possible to tradeoff these

gains for increased range. Figure 6.33 shows three user rate distributions to

illustrate the range extension benefit in going from a baseline MIMO system

to an enhanced MIMO system. The first scenario is the baseline SISO system

operating with reference SNR (which we denote as μ) and cell radius dref .

The second scenario is the MIMO system with improved performance with


the same reference SNR and cell size. The third scenario is the MIMO system

with a reduced reference SNR μ′ < μ and larger cell size d′ref .

User spectral efficiency (bps/Hz)

SISO systemRef SNR μ

Ref SNR μRef SNR μ’

MIMO systemsCD

F

β

R

Fig. 6.33 MIMO can be used to extend range in addition to improving spectral efficiency.

Using an enhanced System B, the reference SNR can be reduced to μ′ to achieve the same

cell-edge performance as the baseline System A at reference SNR μ. For a given transmit

power, the reduced reference SNR results in a larger cell size.

The new reference SNR μ′ is set such that the cell-edge performance of the

MIMO system matches that of the SISO system, for example a cell-edge rate

R corresponding to an outage probability β. For a given transmit power, a

reduced reference SNR corresponds to a network with larger cell radii. Using

the definition of the reference SNR in Section 5.2.2, the extended radius is

d′ref =(μ

μ′

)−γ

dref , (6.10)

where γ is the pathloss exponent. As illustrated in Figure 6.33, the mean user

rate (and hence the mean throughput) of the MIMO system with reference

SNR μ′ will be larger than that of the SISO system. Hence for a given cell-

edge performance level, MIMO can achieve a larger mean and higher range

6.5 Cellular network, coordinated bases 269

compared to SISO. As an alternative to increasing range, one could also use

MIMO enhancements to reduce energy consumption by reducing the transmit

power for a fixed cell size and for a given μ′ < μ.

Conclusion: One can tradeoff the performance gains in spectral effi-

ciency achieved through multiple antennas for increased range (cell size)

or lower transmit power.

6.5 Cellular network, coordinated bases

The results presented so far in this chapter assume that base stations operate

independently, and as a result, the performance is interference limited. In this

section, we consider the performance of networks with coordination between

base stations. As discussed in Section 5.3.2, coordination reduces the intercell

interference but requires higher bandwidth and lower latency on the backhaul

network to share information between bases. We focus on two coordination

techniques: coordinated precoding and network MIMO. The numerical re-

sults presented in the section use the equal-rate criterion for fairness (Section

5.4.4).

6.5.1 Coordinated precoding

Under coordinated precoding (CP), the precoding vectors for each base are

jointly determined using global knowledge of the user channels. With this

knowledge, each base chooses the beamforming weights for its own user(s)

while accounting for the interference it causes to users assigned to other

bases. We consider a downlink coordinated network using minimum sum-

power coordinated precoding, as described in Section 5.4.4.2. Compared to

network MIMO precoding, coordinated precoding is attractive because its

backhaul bandwidth requirements are far more modest.

Our objective is to compare the spectral efficiency performance between a

network with CP and a conventional network with independently operating

bases that use conventional (interference-oblivious) precoding. In the conven-

tional network, each base has CSI for only the users it serves. The simulation


assumptions are summarized in Figure 6.34. Each site is partitioned into

S = 3 sectors, and we fix the number of antennas per sector to be M = 4,

where the sector response is given by (5.7) and Figure 5.5. The number of

antennas per user is N = 1, and we vary K, the number of users per sector,

in the range 1 to 4. We use an equal-rate criterion (but for 10% outage) under

which the beamforming weights and user powers are set to minimize the sum

of the powers transmitted by all sectors while meeting the prescribed SINR

target.


Antennas per sector

Transceiver




Frequency reuse (F)

Transmission mode

Interf.-obliviousprecoding

1 to 4

Coord.precoding

1 to 4

1

30 dB

1

Minimum sum power

3

4

Fig. 6.34 Parameters for system simulations comparing conventional interference-

oblivious precoding with coordinated precoding (Section 6.5.1).

At low reference SNR values, the coordinated and conventional networks

have nearly the same spectral efficiency since the network is noise-limited in

this regime. However, as we go to higher values of the reference SNR, intercell

interference starts to become more significant, and correspondingly, the gains

due to coordination increase. In Figure 6.35, the reference SNR is fixed at 30

dB, a value large enough to make the network interference-limited. It shows

the spectral efficiency (in bps/Hz per site) for CP and conventional precoding

as functions of the number of users per sector.

The spectral efficiency gain due to CP is about 30% with 1 user/sector.

Notice that, in this case, each sector uses up only one spatial dimension

to serve its user, and has three remaining spatial dimensions to mitigate

interference caused to users in other sectors. As the number of users per


1 2 3 40

5

10

15

Number of users per sector

Thro

ughp

ut p

er s

ite (b

ps/H

z)Coordinated precodingInterf.-oblivious precoding

15% gain

Fig. 6.35 Performance of conventional interference-oblivious precoding versus coordi-

nated precoding. Under CP, each base uses global CSI to optimally balance serving as-

signed users and mitigating intercell interference. Despite the additional CSI knowledge,

it achieves only a 15% improvement in throughput.

sector is increased, the gain in spectral efficiency due to CP becomes smaller,

because each sector now has fewer remaining spatial dimensions to mitigate

interference. With 4 users/sector, the gain becomes negligibly small since all

the available spatial dimensions are being used up for MU-MIMO within the

sector.

It is worth noting that the conventional network with K = 4 users/sector

is quite competitive in the best of the CP network cases (corresponding to 2

users/sector), even though the latter has the advantage of global CSI sharing.

This suggests that, in practice, the best complexity-performance tradeoff is

attained when each sector simply uses all the available spatial dimensions

to serve multiple users without regard to the interference caused to users in

other sectors.


Coordinated beam scheduling is a special case of coordinated precod-

ing that shares only scheduling information between bases to avoid beam

collisions. It could improve the cell edge performance over conventional

interference-oblivious precoding, but its throughput performance can be no

better than the coordinated precoding performance and will most likely be

only marginally better than conventional precoding.

Conclusion: Using coordinated base stations for downlink transmis-

sion, if only CSIT (but not user data) is shared among bases, coor-

dinated precoding with global CSIT provides insignificant gains com-

pared to interference-oblivious precoding under single-base operation.

In light of the higher backhaul bandwidth required to share the global

CSIT and the higher complexity to compute the joint precoding weights,

these gains do not justify the costly upgrade. Coordination techniques

that share less information (for example, coordinated scheduling) can

perform no better than coordinated precoding with global CSIT.

6.5.2 Network MIMO

With network MIMO coordination, the antennas at multiple base stations act

like a single array with spatially distributed elements for the transmission and

reception of user signals. As a result, the beamforming weights for each user

can be chosen to mitigate the interference from other co-channel users quite

effectively. Each user is served by a cluster of coordinated bases, and each

base belongs to multiple coordination clusters. Each base has knowledge of

the data signals for all users assigned to those clusters and also has global

knowledge of the channels between all users and all bases. Coordinated pre-

coding is a special case of network MIMO precoding where the coordination

cluster consists of a single base (sector).

Our goal is to understand how the spectral efficiency gain varies with the

number of bases being coordinated (cluster size), the reference SNR, and the

number of antennas per sector. The simulation assumptions are summarized

in Figure 6.36. For the baseline, we assume the site is partitioned into S = 3

sectors and each sector has M = 1, 2 or 4 antennas, with the sector response

given by (5.7) and Figure 5.5. The number of users per sector is M , which,


as we discuss later, is the most favorable case for network MIMO. Each

user is equipped with a single antenna. For the uplink baseline, each user

is detected at a single, optimally assigned base (sector). For the downlink

baseline, each user is served by its optimally assigned base using minimum

sum-power coordinated precoding.

Under network MIMO, the same antenna architectures as the baseline

are used, but antennas are coordinated across each site or across multiple

sites. We consider coordination clusters shown in Figure 5.26 of 1, 7, 19, and

61 sites, which we refer to respectively as 0-ring, 1-ring, 2-ring, and 4-ring

coordination. Because there are up to twelve antennas per site (in the case

of M = 4 antennas per sector), coordination occurs for up to 732 antennas

in the case of 61 sites.

Figure 6.37 shows the uplink spectral efficiency performance under the

equal-rate criterion with different coordination cluster sizes, for M = 1, 2

and 4, respectively. The spectral efficiency, measured in bps/Hz per site, is

plotted as a function of the reference SNR.

Note that, in each case, the spectral efficiency of the conventional system

without any coordination saturates as the reference SNR is increased, indi-

cating that the system is interference-limited. It is this limit that network

MIMO attempts to overcome. The following observations can be made:

• The coordination gain increases with the reference SNR P/σ2 in each

case, because interference mitigation becomes more helpful as the level of

interference between users goes up relative to receiver noise.

• Intrasite coordination provides modest gains in throughput (less than

20%), but these gains can be achieved with only additional baseband pro-

cessing at each site compared to the baseline.

• At the low end of the reference SNR range, most of the spectral efficiency

gain comes just from 1-ring coordination. This is because most of the

interferers that are significant relative to receiver noise are within range

of the first ring of surrounding base stations. However, as reference SNR

is increased, interferers that are further away start to become significant

relative to receiver noise, and therefore it pays to increase the coordination

cluster size correspondingly.

The results from the simulations indicate that, in a high-SNR environment,

the uplink spectral efficiency can potentially be doubled with 1-ring coordi-

nation, and nearly quadrupled with 4-ring coordination. When the user-to-

sector-antenna ratio is smaller than 1, the coordination gain will be somewhat


lower since, even without coordination, each base station can then use the

surplus spatial dimensions to suppress a larger portion of the interference

affecting each user it serves. The coordination gain with a user-to-sector-

antenna ratio larger than 1 will also be lower, because the composite inter-

ference affecting each user at any coordination cluster will then tend towards

being spatially white, making linear MMSE beamforming less effective at

interference suppression.


Antennas per sector

Transceiver




Frequency reuse (F)

Transmission mode

3

Interf.-awareMUD (UL) ,

Coord.precoding (DL)

3 (coordinated)

Network MIMOUL and DL,

up to 4 ringscoordination

1, 2, 4

1

0 to 30 dB

1

Minimum sum power

1, 2, 4

Fig. 6.36 Parameters for system simulations comparing network MIMO with conventional

operation using independent bases (Section 6.5.2).

Figure 6.38 shows the spectral efficiency achievable on the downlink with

different coordination cluster sizes, for M = 1, M = 2, and M = 4, re-

spectively. As on the uplink, the baseline system is interference-limited, and

the same observations apply to the downlink. Also, the gains in spectral effi-

ciency from network MIMO (with different cluster sizes) are quite similar on

the downlink and uplink.

Conclusion: Using coordinated base stations for downlink transmis-

sion, sharing of CSIT and user data allows the possibility of downlink

network MIMO, which implements coherent beamforming across mul-

tiple sites. Uplink network MIMO implements joint coherent detection


Thro

ughp

ut p

er s

ite (b

ps/H

z)

0 10 20 300

10

20

30

40

50

60

Reference SNR (dB)0 10 20 30


Reference SNR (dB)

4-ring coord.2-ring coord.1-ring coord.Intrasite coord.Best sector

4 users/sector4 antennas/sector


1 user/sector1 antenna/sector

Fig. 6.37 Uplink performance using coordinated multiuser detection across multiple sites

(network MIMO). Gains of network MIMO increase as the coordination cluster size in-

creases and as the interference power increases relative to the thermal noise power.

0 10 20 30



Reference SNR (dB)

0

10

20

30

40

50

Thro

ughp

ut p

er s

ite (b

ps/H

z)



1 user/sector1 antenna/sector

4-ring coord.2-ring coord.1-ring coord.Intrasite coord.Best sector

Fig. 6.38 Downlink performance using coordinated precoding across multiple sites (net-

work MIMO). Due to the beamforming duality, the performance trends for the downlink

are similar to those for the uplink.


of the users’ signals across multiple bases. Network MIMO has the po-

tential to increase both mean throughput and cell-edge rates quite dra-

matically when compared to the best MIMO baseline with independent

bases. As the number of coordinated bases increases, the network MIMO

performance improves for a given reference SNR, and the reference SNR

value at which the performance becomes interference limited increases.

In practice, backhaul limitations and imperfect CSI knowledge due to

limited channel coherence reduce the achievable gains.


We have made a number assumptions in our cellular system simulations in

order to simplify the discussion and to focus on fundamental performance

characteristics. In this section, we discuss extensions of the methodology to

account for features and impairments found in real-world networks.

Channel estimation

We have assumed that the channel state information is known perfectly at

the receiver and, when required, at the transmitter. In Section 4.4, we saw

how reference signals are used for channel estimation and how CSI at the

base transmitter is obtained through reciprocity in TDD systems or through

uplink feedback of PMI and CQI in FDD systems. Acquiring more accu-

rate CSI requires a higher investment in power and bandwidth for the pilot

and feedback channels. The overhead also increases when users have higher

mobility or when the channel is more frequency selective.

Overhead for orthogonal pilots increases with the number of antennas (for

common pilots). This is not the case for dedicated pilots because the number

of pilots depends on the number of beams (antenna ports).

We assume that CSI at the receiver (CSIR) can be known ideally by de-

modulating the reference signals. In practice, these estimates will be noisy,

and they can be modeled as additional AWGN [165] [166]. Users with non-

ideal CSIR can therefore be modeled as having reduced geometry.

Modulation and coding

The performance metrics used in our numerical results are based on ideal


Shannon capacity bounds. In practice, the data symbols in our system model

are digitally modulated symbols mapped from channel encoded information

bits. For a given modulation and coding scheme (MCS) indexed by i, the

achievable spectral efficiency (in units of bps/Hz) can be written as

Ti(SNR ) = (1− Fi(SNR ))Ri, (6.11)

where Ri is the spectral efficiency and Fi(SNR ) is the packet error rate (PER)

as a function of SNR at the input of the decoder. The PER can be simulated

offline and is a function of the encoded block length, the coding rate, the

modulation type, and channel characteristics like the mobile speed, and could

incorporate practical impairments like channel estimation error. Plotting the

achievable spectral efficiency using (6.11) for each MCS, the achievable rate

for the overall virtual decoder is given by the outer envelope of the perfor-

mance curves for all MCSs, as shown in the bottom subfigure of Figure 6.39.

The outer envelope is only a few decibels away from the Shannon bound,

and the link performance can be approximated with a 3 dB margin from

the bound: log2(1 + SNR /2). This approximation improves as the number of

MCSs increases. However, more MCSs require additional CQI feedback bits.

Due to constraints on the receiver sensitivity, the modulation is typically

limited to 64-QAM so an upper bound on the achievable rate is 6 bps/Hz.

Therefore the achievable rate in units of bps/Hz can be approximated as

R = min

[6, log2

(1 +

SNR

2

)]. (6.12)

The link performance of various cellular standards is similar because the

encoding is usually based on turbo coding with similar block lengths using

4-QAM, 16-QAM, and 64-QAM.

Wider bandwidths and OFDMA

Our simulations assumed a narrowband channel. In wideband orthogonal

frequency-duplexed multiple-access (OFDMA) channels, frequency selective

fading could be exploited by scheduling resources in both the time and fre-

quency domains. The precoding and rate could be adapted for each uncorre-

lated subchannel, but this would require PMI and CQI feedback for each sub-

channel. A technique known as exponential effective SNR mapping (EESM)

can be used to map a set of channel states across a wide bandwidth into

a single effective SINR that can be used to predict the achievable rate and


-10 -5 0 5 10 15 200

1

2

3

4

5

6

SNR (dB)

Spec

tral e

ffici

ency

(bps

/Hz)

log2(1+SNR/2)

log2(1+SNR)

10-2

10-1

100

-10 -5 0 5 10 15 20SNR (dB)

Pack

et e

rror

rate

64-QAMR=3/4

4-QAMR=3/4

16-QAMR=1/2

4-QAMR=1/2

16-QAMR=3/4

4-QAMR=1/4

4-QAMR=1/8

Fig. 6.39 The top subfigure shows the typical packet error rate performance for a station-

ary user in AWGN. The bottom subfigure shows that the resulting achievable throughput

is well-approximated using the Shannon bound with a 3 dB margin. This approximation

can be used to model the link performance of the virtual decoder.

block-error performance [167]. In this case only a single CQI is used to char-

acterize the wideband SINR.

Because frequency resources can be allocated dynamically across cells un-

der OFDMA, reduced frequency reuse can be implemented selectively for

only users that would benefit most. For example, fractional frequency reuse

as shown in Figure 6.40 deploys reduced frequency reuse for users with low

geometry and universal reuse for all other users. This technique could be

implemented through base station coordination or through distributed al-

gorithms for independent bases [159]. However, as was the case for reduced


frequency reuse (Section 6.3.3), any gains due to fractional frequency reuse

are diminished if multiple antennas are employed at the terminal.

f3

f1

f2

f3

f1f2

f3

f1

f2

: f4

f1 f2 f3 f4

Fig. 6.40 Under fractional frequency reuse, reduced frequency reuse is implemented for

users at the cell edges. Users near the bases are served on the same frequency f4.

Time-varying channels

For high-speed mobiles, the reliability of channel estimation decreases because

the time over which the channel is stationary decreases. Also, the channel

could change significantly between the time the channel is estimated and the

time this information is used for transmission. Techniques that rely on pre-

cise channel knowledge such as CSIT precoding are the most sensitive to the

mismatch. Codebook precoding is more robust, and CDIT precoding is the

most robust because it is based on statistics which vary more slowly than the

CSI.

Limited data buffers

Our simulations assume that each user has an infinite buffer of data to trans-

mit. In more realistic simulations, finite buffers and different traffic models

could be used to represent classes of services such as voice, streaming audio,

streaming video or file transfer.

Synchronization

Coherent transmission and reception requires synchronization so there is no


carrier frequency offset (CFO) among the antenna elements of an array. Co-

located elements belonging to the same site could be connected to a local

oscillator to maintain synchronization. For network MIMO where the ele-

ments are distributed spatially, sufficient synchronization could be achieved

using commercial GPS (global positioning system) satellite signals for out-

door base stations [161]. Alternatively, without relying on GPS, each base

could correct its frequency offset using CFO estimation and feedback from

mobiles [168].

Hybrid automatic repeat request (ARQ)

In an ARQ scheme, an error-detecting code such as a cyclic redundancy check

(CRC) is used to determine if an encoded packet is received error-free. If so,

a positive acknowledgement (ACK) is sent back to the transmitter. If not,

a negative acknowledgement (NAK) is sent and another transmission is sent

with redundant symbols. The process continues until successful decoding is

achieved or until the maximum number of retransmissions occurs. Hybrid

ARQ (HARQ) is a combination of forward error-correcting channel coding

with an error-detecting ARQ scheme. In current cellular standards, turbo

codes or low-density parity-check codes are typically used for channel coding.

When using rate adaptation, there is uncertainty in the SINR knowledge

obtained from CQI feedback due to a number of factors. For example, the

estimates based on the pilots are inherently noisy, the channel could vary over

time, and the interference measured for the CQI feedback could be different

from the interference received during the data transmission. The benefit of

HARQ is that it allows the scheduler to adapt to this uncertainty by using

aggressive rates during the initial transmission followed by retransmissions in

response to NAKs. The simulation results in this chapter cannot benefit from

HARQ because there is essentially no uncertainty in the channel knowledge.

However in practice, HARQ is of great value and is used for almost all data

traffic types including latency-sensitive streaming traffic. An examination of

HARQ in a MIMO cellular network can be found in [169].

6.7 Cellular network design strategies

One can easily be overwhelmed by the choice of MIMO techniques and design

decisions for cellular networks. In this section, we synthesize the insights ac-

6.7 Cellular network design strategies 281

crued over the previous chapters and present a unified system design strategy

for using multiple antennas in macro-cellular networks. This strategy is meant

to provide broad guidelines for deciding among classes of MIMO techniques

and to establish a foundation for more detailed simulations that account for

practical impairments and cost tradeoffs.

Under the assumption of independent bases (Section 5.3.1), the funda-

mental techniques for increasing the spectral efficiency per unit area are as

follows:

1. Increase transmit power. If the transmit powers are set so that the

performance is in the noise-limited regime, the spectral efficiency can be

improved by increasing the powers until the performance becomes interfer-

ence limited (Sections 1.3.1 and 6.3.1). In the interference-limited regime,

increasing the power does not improve the performance significantly.

2. Increase cell site density. By adding cell sites, the throughput per area

scales directly with the cell site density, assuming the number of users per

cell site remains constant (Section 6.3.2). To improve coverage for indoor

environments, indoor bases such as femtocells could be added to offload

traffic.

3. Implement universal frequency reuse. Given reasonable limits on

the transmit power, reusing the spectral resources at each site maximizes

the mean throughput per cell with single-antenna transmission (Section

6.3.3). Fractional frequency reuse provides some gains in the cell-edge per-

formance if the mobiles use a single antenna.

4. Increase spectral efficiency per site. The spectral efficiency could be

increased by improving the link performance, for example with a superior

coding and modulation technique or more efficient hybrid ARQ strategy.

It could also be increased by using multiple antennas to exploit the spatial

dimension of the channel.

These spatial strategies for increasing the spectral efficiency are summarized

in Figure 6.41 and described below. In summary, each site should be par-

titioned into sectors, and MIMO techniques should be implemented within

each sector. Additional performance gains could be achieved by coordinating

bases and implementing network MIMO.

Sectorization

For both the uplink and downlink, the site should be partitioned into sectors,

where the number of sectors is subject to the following three constraints:


1. The angular size of the sector is significantly larger than the angle spread

of the channel.

2. The distribution of users is uniform among the sectors.

3. The physical size of the antennas is tolerable.

Within each sector, the following MIMO techniques for the downlink and

uplink described below could be used.

Downlink MIMO techniques

Because MIMO techniques for the broadcast channel require knowledge of

the CSI at the base station transmitter, the downlink strategies depend on

the reliability of this information. If the CSIT is reliable, for example in

TDD systems (Section 4.4.1), then CSIT precoding (Section 4.2.1) could be

used with either correlated or uncorrelated base antennas. For high-geometry

users, optimal multiplexing gain min(M,KN) can be achieved by using the

multiple antennas to transmit multiple streams to one or more users. For

low-geometry users, serving a single user at a time using TDMA transmission

with a single stream is optimal. Multiple transmit antennas would achieve

precoding gain, and multiple receive antennas would achieve combining gain.

If the CSIT is not reliable, then the MIMO strategy depends on the base

station antenna correlation. Recall that correlation depends on the spacing of

the antennas with respect to the channel angle spread which in turn depends

on the relative height of the antennas compared to the surrounding scatterers.

If the base antennas are correlated, CDIT precoding (Section 4.2.2) or

CDIT precoding (Section 4.2.3) is effective because a small number of precod-

ing vectors could effectively span the subspace of possible channels. For high-

geometry users, multiple users would be served simultaneously to achieve

multiplexing gain. Because the transmit antennas are correlated, the channel

has rank one, and only a single stream is transmitted to each user. For the

case of totally correlated antennas, the multiplexing gain would be no greater

than min(M,K). For low-geometry users, TDMA transmission with a single

stream is optimal, and the multiple antennas provide transmitter and receiver

combining gains.

If the base antennas are not correlated, then the spatial channel would

be too rich for effective codebook/CDIT precoding. If no CSIT is available,

then MU-MIMO techniques cannot be used. Using open-loop SU-MIMO, a

multiplexing gain of min(M,N) can be achieved for high geometry (Section

2.3.3). For low-geometry users, TDMA transmission with a single stream is

6.7 Cellular network design strategies 283

optimal, and multiple receive antennas provide combining gains. Transmit

diversity could provide some gains if there are no other forms of diversity

such as frequency or multiuser diversity.

The multiplexing gains of SU-MIMO for uncorrelated channels, MU pre-

coding for correlated channels, and MU CSIT-precoding are, respectively,

min(M,N), min(M,K), and min(M,KN). For the typical case of N < M <

K , the performance of these techniques increases roughly in this order, as

shown in Figure 6.41. For SU transmission, the rate improves linearly as N in-

creases for high-geometry users due to multiplexing gain (the rate improves

linearly as long as N ≤ M) and for low-geometry users due to combining

gain (linear gains in post-combining SNR result in linear rate gains at low

geometry). For MU transmission, increasing the number of mobile antennas

N provides combining gain but does not increase the multiplexing gain for

MU transmission if K > M . At high geometry, combining gain results in

a logarithmic rate increase, but at low geometry, it results in a linear rate

increase. Therefore increasing N is beneficial mainly for cell-edge users under

MU transmission.

Uplink MIMO techniques

The multiplexing gain of the ((K,N),M) MAC is min(M,KN). For a typ-

ical scenario where the number of users exceeds the number of base station

antennas (K < M), full multiplexing gain can be achieved by transmitting

a single stream from each user, and in this case CSIT is not required by the

users. Therefore uplink MIMO performance with full multiplexing gain can

be achieved with only CSI at the receiver. The transceiver options for the

uplink are simpler than for the downlink because CSI estimates are easier

to obtain at the receiver than at the transmitter. If the CSIR is reliable,

then multiple users should transmit simultaneously within each sector and

capacity-achieving MMSE-SIC could be used. If the CSIR is not reliable, then

the more robust MMSE receiver could be used. Multiple terminal antennas

could provide precoding gain, but this would require CSIT which could be

difficult to obtain in practice.

Coordinated bases

By coordinating the bases to share global CSIT, coordinated downlink pre-

coding could be used to mitigate intercell interference. However, the gains are

minimal compared to conventional interference-oblivious transmission with

independent bases, and these gains do not justify the cost of acquiring global


CSIT. Additional gains could be achieved using interference alignment, but

its performance has not been evaluated in this book.

If the backhaul could be enhanced so that bases share both global CSIT

and user data, then downlink coordinated transmission with network MIMO

and CSIT precoding could reduce intercell interference to provide more sig-

nificant gains compared to the baseline. Uplink network MIMO performs

joint detection across multiple bases and could provide similar gains. The

performance gains of network MIMO increase as the number of coordinated

bases increases. However, increasing the coordination cluster size requires ad-

ditional backhaul resources and acquiring accurate CSI among all transmitter

and receiver pairs becomes more difficult.

Uncorrelated base antennas

Correlated or uncorrelatedbase antennas

ReliableCSIT?

Correlated base antennas?no

no

yes

MU-MIMO: CSIT precoding

SU-MIMO

yes

Correlated or uncorrelatedbase antennas

Downlink

Uplink

MU-MIMO: MMSE-SIC Add network MIMO

MU-MIMO:Codebook/CDIT precoding

Correlated base antennas

* Subject to requirements on 1. angle spread 2. antenna size 3. user distribution

Add network MIMO

Improving performance

Partition site into sectors*

Partition site into sectors*

Fig. 6.41 Overview of design recommendations for sectorization and MIMO techniques

to increase spectral efficiency per site in a cellular network.

Chapter 7

MIMO in Cellular Standards

Standardization of cellular network technology is required to ensure interop-

erability between base stations, as well as handsets manufactured by different

companies. MIMO standardization at the physical layer focuses on the trans-

mission signaling, describing how information bits are processed for trans-

mission over multiple antennas. Receiver techniques are determined by the

terminal or base station vendor and do not need to be standardized.

Since the early 2000s, MIMO techniques have been adopted in cellular

standards in parallel with the development of the MIMO theory. The earliest

MIMO standardization focused on downlink single-user spatial multiplexing

to address the demand for data downloading and higher peak data rates.

Recently, there has been more interest in uplink MIMO, multiuser MIMO,

and coordinated base techniques. Higher-order sectorization based on fixed

sectors does not require standardization because the physical layer is defined

for a given sector and is independent of the number of sectors per site.

This chapter gives an overview of conceptual aspects of MIMO techniques

for two families of standards. One is based on the Third Generation Part-

nership Program (3GPP) specification and includes the Universal Mobile

Telecommunications Systems (UMTS), Long-Term Evolution (LTE), and

LTE-Advanced (LTE-A) standards. The 3GPP specification evolves through

a series of releases. MIMO techniques were first introduced in UMTS as part

of the 3GPP Release 7 specification. LTE is defined in Releases 8 and 9,

and LTE-A is defined in Release 10. The other family of cellular standards

is based on the IEEE 802.16 specification for broadband wireless access in

metropolitan area networks. The IEEE 802.16e standard addresses mobile

wireless access in the range of 2 GHz to 6 GHz and is often known as mobile

WiMAX (Worldwide Interoperability for Microwave Access, a forum that de-


2857

286 7 MIMO in Cellular Standards

fines subsets of the 802.16e standard for commercialization). IEEE 802.16m

is the evolution of IEEE 802.16e and is sometimes known as Mobile WiMAX

Release 2. Both LTE-A and IEEE 802.16m are designed to achieve the Inter-

national Telecommunication Union (ITU) requirements for fourth-generation

cellular standards which include an all-IP packet switched network architec-

ture, peak rates of up to 1 Gbps for low mobility users, and mean throughput

per sector of up to 3 bps/Hz [170].

Figure 7.1 summarizes the key features of these standards. Additional

details can be found in books [4] [171] [5], tutorial articles [172] [173] [174]

[175] [176] and the actual standards documents.

UMTS

5 MHzBandwidth

DL

SU-MIMO

MU-MIMO

LTE

Up to 20 MHz

LTE-A

Up to 100 MHz

802.16e

Up to 20 MHz

802.16m

Up to 100 MHz

CDMA

2 streams

Multipleaccess

SU-MIMO

MU-MIMO

DL: OFDMAUL: SC-FDMA

4 streams

4 users

DL: OFDMAUL: SC-FDMA

4 streams

8 users4 users

OFDMA

8 streams

4 users

4 streams

8 users

OFDMA

2 streams

UL

8 streams

4 users

Fig. 7.1 Summary of MIMO options for cellular standards. The maximum number of

multiplexed streams and users are shown for SU-MIMO and MU-MIMO, respectively.

7.1 UMTS

Single-user spatial multiplexing was first adopted in cellular networks in the

UMTS standard for high-speed downlink packet access (HSDPA). The UMTS

standard is based on code-division multiple access (CDMA) in 5 MHz band-

width, and the HSDPA extension achieves high data rates over the high-speed

downlink shared channel (HS-DSCH) by transmitting data on up to 15 out of

16 orthogonal spreading codes. The multiple antenna techniques are defined

for up to two base station antennas.

HSDPA supports open-loop transmit diversity, known as space-time trans-

mit diversity (STTD). A similar technique termed space-time spreading (STS)

[177] was also adopted as an optional transmit mode for circuit-switched voice

7.1 UMTS 287

in 3GPP2 in 1999 [178]. A block diagram is shown in the top half of Fig-

ure 7.2. Information bits are channel- encoded, interleaved, and mapped to

QPSK or 16-QAM constellation symbols. The data symbols are multiplexed

into multiple substreams (up to 15) which are modulated using mutually or-

thogonal CDMA spreading codes. These signals are summed and modulated

using a cell-specific scrambling sequence. Signals from pairs of consecutive

symbol intervals are modulated using Alamouti space-time block coding and

transmitted over the two antennas. Because STTD does not depend on the

channel realizations, it can be used for terminals with high mobility.

If the terminal is moving slowly, closed-loop transmit diversity and spatial

multiplexing can be used to exploit feedback from the terminal. Release 7 of

the UMTS standard supports up to two spatially multiplexed streams using

up to 64-QAM modulation. While the standard defines the modulation and

coding to achieve a peak data rate of 42 Mbps (using two streams with 64-

QAM and 0.98 rate coding), this rate is achievable in practice with very low

probability, for example, with stationary users located directly in front of

the serving base. While aggressive values for peak rate are often defined in

standards, they are not necessarily achievable in realistic environments or

with significant probability.

Spatial multiplexing is accomplished through precoding, as shown in Fig-

ure 7.2. Information bits are first multiplexed into two streams that are sep-

arately encoded, interleaved, and mapped to constellation symbols. The two

streams have the same modulation and coding, and they are also spread and

scrambled using the same sequences. The streams are precoded using a 2× 2

matrix G drawn from a codebook of four unitary matrices:{[12

12

1+j

2√2

−(1+j)

2√2

],

[12

12

1−j

2√2

−(1−j)

2√2

],

[12

12

−1+j

2√2

−(−1+j)

2√2

],

[12

12

−1−j

2√2

−(−1−j)

2√2

]}.

(7.1)

HSDPA spatial multiplexing is sometimes known as dual-stream transmit

adaptive array (D-TxAA) transmission and is a generalization of the closed-

loop transmit diversity technique known as TxAA. Under TxAA, a sin-

gle stream is precoded using the weights given by the (re-normalized) left

columns of the SM precoding matrices:{[1√2

1+j2

],

[1√2

1−j2

],

[1√2

−1+j2

],

[1√2

−1−j2

]}. (7.2)


Precoding

Spreading, scrambling

CC+I+M

Closed-loop spatial multiplexing

Open-loop transmit diversity

Spreading,scrambling

Spreading, scrambling

Coding,modulation

Coding,modulation

MuxData bits

Data bits Coding,

modulation STBC

Fig. 7.2 Block diagrams for UMTS MIMO transmission. Up to M = 2 base station

antennas are supported.

The mobile feeds back precoding matrix indicator (PMI) bits to indicate

its preferred weights (Section 5.5.3). Because the codebooks each have four

entries, two bits are required for the PMI. We note that SM precoding for two

transmit antennas and two streams requires very little PMI feedback, because

given one precoding vector, its companion orthogonal vector is immediately

known. (A note on terminology: the indicator bits in the UMTS standard are

called precoding control information (PMI) bits.)

Under spatial multiplexing, the received signal will suffer from inter-code

interference as a result of frequency-selective fading and from inter-stream

interference as a result of non-ideal unitary precoding. The receiver can ac-

count for both effects using multiuser detection techniques combined with

a front-end rake receiver or equalizer. A survey of HSDPA receiver designs

based on front-end equalization can be found in [179].

7.2 LTE

The Long-Term Evolution (LTE) standard is the next-generation evolution of

the UMTS standard based on a downlink orthogonal frequency division multi-

ple access (OFDMA) and an uplink single-carrier frequency-division multiple-

access (SC-FDMA) air interface. Compared to UMTS, higher data rates are

achieved through wider bandwidths (up to 20 MHz compared to 5 MHz)

7.2 LTE 289

and higher order spatial multiplexing (up to four streams compared to two).

LTE supports up to four antennas at the base. LTE also offers a wider vari-

ety of MIMO transmission options including open-loop spatial multiplexing,

generalized beamforming, and MU-MIMO.

Freq resource mapping+IFFT


NALayermapping

NL

Mobile feedback, closed-loopmode

Mobile feedback, open-loopmode

RI CQI (per stream)

PMI

RI CQI (same for

each stream)

Data bits

Precoding

Coding,modulation

Coding,modulation

Mux �

Fig. 7.3 Block diagram for downlink LTE MIMO transmission. Up to M = 4 base sta-

tion antennas are supported. Different precoding is used for closed-loop and open-loop

techniques.

On the downlink, three types of single-user transmission are supported:

closed-loop transmission (both spatial multiplexing and transmit diversity),

open-loop transmission (both spatial multiplexing and transmit diversity)

and generalized beamforming. Closed-loop transmission is more suitable for

terminals with low mobility and requires more feedback from the terminals

than open-loop transmission. LTE also has minimal support for MU-MIMO

on both the downlink and uplink.

Downlink SU closed-loop transmission

The transmitter block diagram for LTE downlink spatial multiplexing is

shown in Figure 7.3. Information bits are multiplexed into two streams, and

each stream is independently coded, interleaved, and mapped to constellation

symbols. (In the LTE standard, the term “codeword” is used to denote the

independently encoded data streams. To avoid confusion with the precod-

ing codewords, we use the term “stream” instead.) Cell-specific scrambling

occurs between the interleaving and modulation to ensure interference ran-

domization between cells. The modulation and coding for each stream can be


different and could be based on the CQI feedback from the mobile (Section

5.5.3). If there are four transmit antennas, the two streams are mapped to NL

layers (NL = 2, 3, 4) that correspond to the rank of the spatially multiplexed

transmission. The number of layers is suggested by the rank indicator (RI)

feedback from the mobile. If there are two transmit antennas, the number of

layers is NL = 2.

The precoding is implemented with a NA × NL matrix, where NA is the

number of antennas (known as ports in the standard). The precoding can

be implemented in a frequency-selective manner where different weights are

applied across different subbands, or it can be implemented with a single set

of weights for the entire transmission band. The precoding is based on the

PMI feedback from the mobile. For NA = 2, one of three precoding matrices

are used:

G ∈{[

1√2

0

0 1√2

],

[12

12

12

−12

],

[12

12

j2

−j2

]}. (7.3)

The entries for the precoding matrices have been designed so that no complex

multiplications are required, only phase rotations by 90, 180, and 270 degrees.

For NA = 4, the precoding matrices are chosen from a codebook of size 16.

The 4×NL matrix corresponds to the NL columns of Householder matrices

given by:

Gn = I4 − 2unu

Hn

uHn un

, (7.4)

where n (n = 0, ..., 15) is the codebook index, and un is a unique 4 × 1

generating vector.

After precoding, the data symbols are distributed across the frequency-

domain resources to provide frequency diversity. Closed-loop transmit diver-

sity is suitable when the mobile has low mobility and sufficiently high SINR

to support multiple streams. If the mobile has low mobility but lower SINR,

then closed-loop transmit diversity can be used.

For closed-loop transmit diversity, only a single stream is encoded, and

this stream is mapped to a single layer. For NA = 2 antennas, the precoding

vector is chosen from a codebook of size 4:

g ∈{[

1√21√2

],

[1√21

−√2

],

[1√2j√2

],

[1√2

−j√2

]}. (7.5)

Note that HSDPA also used four vectors but with different values. Closed-

loop transmit diversity is sometimes known as codebook-based beamforming.

7.2 LTE 291

For NA = 4 antennas, one of 16 precoding vectors are used, and these vectors

are given by the first column of the Householder matrices (7.4).

Downlink SU open-loop transmission

Open-loop spatial multiplexing, sometimes known as large-delay cyclic delay

diversity (CDD), is well-suited for high-mobility users whose precoding feed-

back is not reliable. All layers are constrained to be the same rate, so only a

single CQI feedback value is required. After the layer mapping, the layers are

precoded to provide diversity across the antennas with the effect of averaging

the SINR of each layer. Different precoding matrices are used for frequency

different subbands to provide additional frequency diversity.

Open-loop transmit diversity is achieved using Alamouti block coding over

the frequency domain. With two transmit antennas, the antenna mapping

block-encodes consecutive symbols across consecutive subcarriers, as shown

in Figure 7.4. This is known as space-frequency block coding (SFBC). With

four transmit antennas, the block encoding is similar, but the transmission

occurs over alternating antenna pairs to provide additional spatial diversity.

This technique is known as SFBC with frequency shift transmit diversity

(FSTD).

Given a sequence of encoded symbols{b(t)}

(t = 1, 2, . . .), each pair of

symbols on successive time intervals b(2j−1) and b(2j) (j = 1, 2, . . .) is trans-

mitted over the 2 antennas on intervals 2j − 1 and 2j as follows:

s(2j−1) =

[b(2j−1)

b(2j)

]and s(2j) =

[−b∗(2j)

b∗(2j−1)

].

Downlink SU generalized beamforming

If the base station antennas are closely spaced and the channel angle spread

is narrow, then directional beamforming can be used to extend the transmis-

sion range or to provide transmit combining gain. The beamforming weights

in this case would not be drawn from a codebook but would be matched to

the direction of the intended user. These non-codebook-based weights could

be obtained in an FDD system via estimation of the second order statistics

of the uplink channel. Given the beamforming vector g ∈ CM×1, dedicated

reference signals (Section 4.4) are transmitted using g in order for the ter-

minal to estimate the channel hHg. In using the dedicated reference signals,

the mobile does not need to have knowledge of g. Generalized beamforming


Onesubcarrier

Empty resource element

Port 0

Port 1

Port 0

Port 1

Port 2

Port 3

Fig. 7.4 LTE frame structure for open-loop transmit diversity using space-frequency block

coding.

supports transmission to only a single user so no multiplexing gains can be

achieved.

Downlink MU-MIMO

LTE Release 8 supports beamforming for two users (one layer per user) over

the same time-frequency resources. The precoding vectors are drawn from

the codebooks for closed-loop transmit diversity. Common reference signals

(Section 4.4) are used, and the CQI and PMI feedback sent by each user is

computed assuming that it is the only user served. The actual achievable rate

would be lower than the predicted rate as a result of interference that occurs

during the data transmission phase. Interference is due to interbeam inter-

ference from the other user scheduled by the base and intercell interference.

The intercell interference could be especially detrimental if the user lies in

the direction of an interfering beam, and the potentially significant reduction

of SINR is known as the flashlight effect.

For LTE Release 9, dedicated reference signals are defined for two layers,

enabling non-codebook-based precoding such as zero-forcing for two users.

Even though the precoding vectors do not belong to the codebook, the PMI

feedback is based on the codebook.

Uplink transmission

On the uplink, SU-MIMO multiplexing is not supported. However, antenna

selection is supported for terminals with multiple antennas. In the case of

multiuser transmission, a base can schedule more than one user (up to eight)

7.3 LTE-Advanced 293

on a given time-frequency resource using SDMA. Users are distinguished

from one another using orthogonal reference signals which are transmitted

on every fourth time-domain slot.

Two types of reference signal are used for uplink transmission. Demodu-

lation reference signals (DRS) are used during data transmission to enable

coherent detection. Sounding reference signals (SRS) are transmitted by the

terminal for the purpose of resource allocation and estimating the SINR.

To enable frequency selective resource allocation, the SRS cover the entire

bandwidth during a single transmission interval or by hopping over different

subbands over multiple time intervals. In contrast, the DRS are transmitted

on only the subbands that carry user data.

7.3 LTE-Advanced

The LTE-Advanced system uses wider bandwidths (up to 100 MHz) and

higher multiplexing order for downlink SU-MIMO. In addition, it allows up-

link SU-MIMO, and coordinated base station techniques, known as coordi-

nated multipoint (CoMP) reception and transmission, are under considera-

tion. At the system level, LTE-A can provide higher spectral efficiency per

unit area using heterogenous networks that mix macrocells, femtocells, and

relays. LTE-A supports up to eight antennas at the base station with array

architectures following the four-antenna configurations in Figure 5.27: eight

closely spaced V-pol columns, four columns of closely spaced X-pol columns,

and four columns of diversity spaced X-pol columns.

Downlink transmission

On the downlink, higher peak data rates (up to 1 Gbps) can be achieved

by leveraging the wider bandwidth and using up to eight layers (with two

codewords) for closed-loop SU-MIMO transmission. Of course eight antennas

would be required at the base, and at least eight antennas would be required

at the terminal. Higher throughput can be achieved using closed-loop MU-

MIMO with non-codebook-based precoding for up to four layers and up to

two layers per user. For example, the base can serve four users each with one

layer or two users each with two layers. LTE-A supports dynamic switching


between the SU and MU modes, so the base can decide from frame to frame

how to distribute its layers among the users.

Codebooks for two and four transmit antennas are the same as those de-

fined for LTE. For eight transmit antennas, a dual-codebook technique is

used such that each codeword is the product of two codewords drawn from

two different codebooks. One codebook is designed for the wideband, long-

term spatial correlation characteristics. The other codebook is designed for

the short-term, frequency-selective properties of the channel. PMI feedback is

sent independently for the two codebooks, with the rates scaled accordingly

so the latter type occurs more frequently.

Downlink LTE-A supports both common and dedicated reference signals

for up to eight transmit antennas. The common reference signals are known

as channel state information reference signals (CSI-RS). To facilitate CoMP

transmission over many bases, up to 40 CSI-RS are defined so that, for exam-

ple with eight antennas per sector, a frequency reuse of 1/5 could be used to

differentiate channels from up to five sectors. The demodulation reference sig-

nals (DRS) are dedicated reference signals that are modulated using the pre-

coding vector(s). They are used to support non-codebook-based MU-MIMO

precoding and can be also be used for codebook-based SU-MIMO precoding.

Because there is a DRS associated with each precoded layer, whereas there is

a CSI-RS associated with each antenna, channel estimation is more efficient

using DRS (for fixed reference signal resources) if the number of layers is less

than the number of antennas.

Different types of downlink CoMP transmission strategies are under con-

sideration. Joint processing CoMP is closely related to network MIMO and

uses DRS that are jointly precoded across clusters of coordinated bases. Co-

ordinated beam scheduling is also under consideration. In light of the per-

formance results in Chapter 6, minimal gains in throughput compared to

conventional single-base precoding would be expected.

Uplink transmission

In addition to supporting uplink MU-MIMO for backwards compatibility

with LTE, LTE-A also supports SU-MIMO transmission for two and four

antennas. Up to four layers are supported for four antennas. Codebook-based

precoding is used for the transmission, deploying the same mechanism of

PMI, RI, and CQI feedback from the base to the mobile. However, different

codebooks (not based on Householder matrices) are used in order to minimize

the peak-to-average power ratio [172]. This is accomplished by allowing each

7.4 IEEE 802.16e and IEEE 802.16m 295

antenna to transmit at most a single layer. The DRS are precoded using the

same vectors as the data layers. On the other hand, the SRS are transmitted

from each antenna.

CoMP reception on the uplink using network MIMO can be accomplished

as a vendor-specific feature and does not require any enhancements in the

air-interface standard.

7.4 IEEE 802.16e and IEEE 802.16m

The IEEE 802.16e standard is based on an OFDMA air interface for both the

uplink and downlink. Like LTE, it can operate with variable bandwidths from

1.25 to 20 MHz by scaling the FFT size. 802.16e supports both open-loop

and closed-loop transmission for two base antennas.

The closed-loop techniques are for single-stream transmission that rely on

TDD channel reciprocity for determining precoding weights. Under maximal

ratio transmission, the precoding weights are computed to maximize the SNR

individually on each subcarrier. The channel is assumed to change slowly

enough in time so that reliable uplink channel estimates can be made on a

per-subcarrier basis. For more rapidly changing channels, a single precoding

vector can be applied across the entire band for all subcarriers using statistical

eigenbeamforming as a form of CDI precoding. In this case, the precoding

vector is a function of the largest eigenvector of the channel covariance matrix.

802.16e also supports open-loop transmission over two base antennas. The

block diagram is shown in Figure 7.5 for transmit diversity and spatial multi-

plexing, known respectively as Matrix A and Matrix B transmission. Transmit

diversity is achieved using Alamouti space-time block coding. Spatial multi-

plexing in this case occurs after the coding and modulation, and this option

is known as vertical encoding. As was the case for open-loop SU-MIMO for

LTE, CQI feedback is required for only a single data stream.

IEEE 802.16m is a future enhancement of 802.16e with additional MIMO

schemes and with support for up to eight base station antennas and four

mobile antennas. Both open and closed-loop SU-MIMO are based on vertical

encoding and support up to eight streams on the downlink and up to four

streams on the uplink. Downlink MU-MIMO supports up to four users and up

to a total of four streams using codebook or CSIT precoding. CDIT precod-

ing can also be implemented using uplink feedback of the downlink channel


Matrix A or B

Data bits

Coding,modulation

Mux



Fig. 7.5 Block diagram for 802.16e open-loop transmission. Matrix A is for Alamouti

space-time block coding, and Matrix B is for spatial multiplexing.

covariances to generate a codebook of beamforming vectors that span the

likely directions of users. This codebook can be adapted based on additional

covariance feedback. If the mobile is slowly moving, then its desired precoding

vector is correlated in time. To exploit this correlation, IEEE 802.16m intro-

duces differential codebooks so the PMI feedback denotes the incremental

change between the previous and current desired codeword.

The IEEE 802.16m standard also supports two types of coordinated beam

scheduling. If a base knows the direction of an active user, it can notify an

adjacent base of restricted precoding vectors that would cause interference.

Conversely, a base can notify an adjacent base of recommended precoding

vectors that would not cause interference. These strategies are known re-

spectively as PMI restriction and PMI recommendation. Network MIMO

coordinated precoding, for example based on zero-forcing, is also supported.

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176. K. Balachandran, D. Calin, F.-C. Cheng, N. Joshi, J. H. Kang, A. Kogiantis,

K. Rausch, A. Rudrapatna, J. P. Seymour, and J. Sun, “Design and analysis of an

IEEE 802.16e-based OFDMA communication system,” Bell Labs Technical Journal,

vol. 11, no. 4, pp. 53–73, 2007.

177. B. Hochwald, T. Marzetta, and C. B. Papadias, “A transmitter diversity scheme for

wideband CDMA systems based on space-time spreading,” IEEE Journal on Selected

Areas in Communications, vol. 19, no. 1, pp. 48–60, Jan. 2001.

178. A. Kogiantis, R. Soni, B. Hochwald, and C. Papadias, “Downlink improvement

through space-time spreading,” IS-2000 standard contribution 3GPP2-C30-19990817-

014, Aug. 1999.

179. S. P. Shenoy, I. Ghauri, and D. Slock, “Receiver designs for MIMO HSDPA,” in

Proceedings of the IEEE International Communications Conference, 2008.

Index

3rd Generation Partnership Project

(3GPP), 285

Alamouti space-time block coding, see

space-time block coding

angle spread, 217

antenna array architecture, 21–22, 213–214

closely-spaced, 9, 22, 214–216

cross-polarized (X-pol), 213

diversity-spaced, 214, 216

for sectors, 215–217

multibeam, 216

omni-directional, 167

omnidirecitonal element, 21

uniform linear array (ULA), 214, 216

vertically-polarized (V-pol), 213

antenna port, 290

backhaul, 162, 224, 225

base station, 1, 163, 229

base station coordination, 31, 181–186, 205

backhaul, 284

centralized architecture, 224

coordinated precoding, 181–182, 210,

269–272, 283

distributed architecture, 224

interference alignment, 182–183, 284

network MIMO, 184–186, 210, 272–274

beamforming, 71

directional, 22, 72–73, 144, 214, 216, 228,

263–267, 291

suboptimal technique for the MAC, 122

block diagonalization (BD) precoding, 133

block fading channel model, 37

broadcast channel (BC)

capacity region, 12, 15–17, 87–93

capacity-achieving technique, 16, 93

model, 5, 81

sum-rate capacity performance

versus the number of base antennas,

113

versus the number of users, 114–115

compared to SU-MIMO, 103–104

high SNR, 17, 18, 106

low SNR, 16, 19, 105

weighted sum-rate, 99–101

Butler matrix, 216

capacity

closed-loop MIMO channel, 10, 45–48,

53–54

multiple-input single-output (MISO)

channel, 8, 53–54

open-loop MIMO channel, 43–45, 53–54

single-input multiple-output (SIMO)

channel, 7, 53–54

single-input single-output (SISO)

channel, 6

sum-rate, 12

capacity region, 12

broadcast channel (BC), 12, 15–17,

87–93

309

310 Index

multiple-access channel (MAC), 12–15,

81–87

carrier frequency offset (CFO), 225, 279

CDIT precoding, see channel distribution

information at the transmitter

precoding

CDMA, see code division multiple access

cell site, 21, 160

cell-edge user rate, 195

cellular network, 1, 19–31

angle spread, 217

antenna array architecture, 213–214

base station, 1, 163, 229

base station coordination, 31, 181–186,

205, 210, 223–225, 283, 284

design components, 160–164

downlink channel model, 165

frequency reuse, 21, 238–243, 281

interference mitigation, 29–31, 284

pathloss exponent, 166, 268

performance

coordinated bases, 269–276

interference-limited, 28, 227, 234, 270,

273, 274, 281

isolated cell, 229–232

multiple-antenna bases, 249–269

noise-limited, 28, 233, 270, 281

single-antenna bases, 232–249

performance criteria, 188–190, 229

performance metric, 195

reference distance, 166

reference SNR, 169–174, 228

scheduling, 24–26

sectorization, 19–22, 160, 167–169, 228,

243–249, 253–267, 281

shadowing, 27, 166

simulation methodology, 186–213

terminal, 1, 163, 229

uplink channel model, 164

wraparound, 174–176, 228

centralized base station coordination

architecture, 224

channel distribution at the transmitter

(CDIT) precoding, 142

channel distribution information at the

transmitter (CDIT) precoding,

141–295

channel model type

block fading, 37

double-bounce, 40

fast fading, 37

Hata, 166

i.i.d. Rayleigh, 4, 37, 228

keyhole, 40

Kronecker model, 38

line-of-sight (LOS), 9, 144

physical, 41

Ricean, 41

single-bounce, 39

spatially rich, 4, 37

Weichselberger, 40

channel quality indicator (CQI), 218, 220,

222–223, 277, 290–292, 294, 295

channel state information (CSI)

estimation, 76, 117, 276, 283

channel state information at the trans-

mitter (CSIT), 222–223, 282,

283

channel state information at the transmit-

ter (CSIT) precoding, 128–141, 220,

222, 279, 282, 284, 295

channel state information reference signal

(CSI-RS), 294

closely-spaced antenna elements, 9, 22,

214–216

co-channel interference, 26–27

code division multiple access (CDMA), 22,

286

codebook precoding, 70–76, 142–146, 218,

287, 290, 294, 295

codeword, 71

common reference signal, 292, 294

coodination cluster, 184

coordinated multipoint (CoMP), 293–295

coordinated precoding, 181–182, 210, 296

coordination cluster, 205, 272

cross-polarized (X-pol) antenna elements,

213

Index 311

CSIT precoding, see channel state

information at the transmitter

precoding

cyclic delay diversity (CDD), 291

dedicated reference signal, 294

degraded broadcast channel, 89

demodulation reference signal (DRS), 293,

294

diagonal BLAST (D-BLAST), 66–67

directional beamforming, 22, 214, 216

system performance, 263–267

dirty paper coding (DPC), 16, 89, 91, 93,

118, 194

distributed base station coordination

architecture, 224

diversity-spaced antenna elements, 214,

216

Dolph-Chebyshev array response, 144, 263,

265

downlink cellular channel model, 165

dual-stream transmit adaptive array

(D-TxAA), 287

duality

linear transceivers, 149–153, 208

multiple-access and broadcast channel

(MAC-BC), 93–96, 99

eigenmode, 10

equal-rate performance criteria, 189, 229

equal-rate system simulation methodology,

202–213

exponential effective SNR mapping

(EESM), 278

fast fading channel model, 37

fast Fourier transform (FFT), 24

flashlight effect, 292

fractional frequency reuse, 278

frequency division multiple access

(FDMA), 22

frequency reuse, 21

fractional, 278

system performance, 238–243, 281

frequency shift transmit diversity (FSTD),

291

frequency-division duplexed (FDD) system,

162, 181, 222, 276, 291

geometry, 27, 188, 197

global positioning system (GPS), 225, 280

Hata channel model, 166

High-speed downlink packet access

(HSDPA), 286

High-speed downlink shared channel

(HS-DSCH), 286

higher-order sectorization, 168, 254, 261,

262

HSDPA, see High-speed downlink packet

access

hybrid ARQ (HARQ), 280, 281

i.i.d. Rayleigh channel, 4, 37, 228

IEEE 802.16e, 24, 285, 295

IEEE 802.16m, 214, 286, 295–296

interference alignment, 182–183

interference mitigation, 29–31

interference-aware, 179–181

interference-oblivious, 178–179, 242, 262,

269, 272, 284

interference-limited system performance,

28, 227, 234, 270, 273, 274, 281

International Telecommunication Union

(ITU), 286

line-of-sight (LOS) channel, 9

Long-Term Evolution (LTE), 24, 218, 285,

288–293

LTE-Advanced (LTE-A), 214, 285, 293–295

matched filter (MF) receiver, 59

maximal ratio combiner (MRC), 7, 59

maximal ratio transmitter (MRT), 9, 68,

144

maximum sum rate performance criteria,

188

mean throughput per cell site, 195

medium access control (MAC) layer, 163

MIMO channel

capacity performance, 53–54

high SNR, 11, 18, 51–52

312 Index

large number of antennas, 52–53

low SNR, 11, 19, 49–51

closed-loop capacity, 10, 45–48

closed-loop capacity-achieving technique,

68

model, 3, 35–37, 43

open-loop capacity, 43–45

open-loop capacity-achieving technique,

65, 66

minimum mean-squared error (MMSE)

receiver, 60, 124

MISO, see multiple-input single-output

capacity

MMSE with successive interference

cancellation (MMSE-SIC) receiver,

14, 61, 63, 84, 86, 124, 283

modulation and coding scheme (MCS), 277

multibeam antenna, 216

multiple-access channel (MAC)

capacity region, 12–15, 81–87

capacity-achieving technique, 14, 86

model, 4, 79

sum-rate capacity performance

versus the number of base antennas,

109

versus the number of users, 115

compared to SU-MIMO, 103–104

high SNR, 15, 18, 106

low SNR, 14, 19, 104


multiple-input single-output (MISO)

capacity, 8

multiuser detection, 14, 124

multiuser diversity, 221

multiuser eigenmode transmission (MET)

precoding, 135–141

multiuser MIMO, see broadcast channel

and multiple-access channel

network MIMO, 184–186

system performance, 272–274

next-generation mobile networking

(NGMN) methodology, 166, 173, 174,

227

noise-limited system performance, 28, 233,

270, 281

non-degraded broadcast channel, 91

OFDMA, see orthogonal frequency division

multiple access

omni-directional antenna element, 21, 167

orthogonal frequency division multiple

access (OFDMA), 22, 277, 288

pathloss exponent, 166, 268

peak user rate, 195

per-antenna power constraint, 132

per-antenna rate control (PARC), 63

per-user unitary rate control (PU2RC),

143

physical (PHY) layer, 19, 163

physical resource block (PRB), 24

pilot signal, see reference signal

power azimuth spectrum (PAS), 169

power gain, 8, 18

pre-log factor, see spatial multiplexing gain

precoding, 9

multiuser eigenmode transmission

(MET), 135–141

block diagonalization (BD), 133

channel distribution at the transmitter

(CDIT), 295

channel distribution information at the

transmitter (CDIT), 141–142, 216,

223, 279, 282

channel state information at the

transmitter (CSIT), 128–141, 220,

222, 279, 282, 284, 295

codebook, 70–76, 142–146, 218, 295

maximal ratio transmitter (MRT), 9, 68,

144

multiple users, 125–148

random, 146–148

single-user, 70–76

zero-forcing (ZF), 129–141, 145

precoding control information (PCI), 288

precoding matrix indicator (PMI), 218,

220, 222–223, 277, 288, 290, 292, 294,

296

Index 313

proportional-fair performance criteria, 188,

190, 229

proportional-fair scheduling, 191

proportional-fair system simulation

methodology, 194–201

quality-of-service (QoS) weight, 95, 190,

220

random precoding, 146–148

rank indicator (RI), 290

rate adaptation, 25

receiver architecture

matched filter (MF), 59

maximal ratio combiner (MRC), 7, 59

minimum mean-squared error (MMSE),

60, 124

MMSE with successive interference

cancellation (MMSE-SIC), 14, 61, 63,

84, 86, 124

reference distance, 166

reference signal

common, 276

dedicated, 276

reference SNR, 169–174, 228

scheduling, 219–221

proportional-fair, 191

SCM, see spatial channel model sector

antenna response

sector antenna, 215–217

sectorization, 160, 167–169, 228

higher-order, 216, 217

Dolph-Chebyshev array response, 144,

263

higher-order, 168, 243–249, 253–267

ideal response, 167

parabolic response, 168

spatial channel model (SCM), 168

system performance, 243–249, 256, 262,

281

shadowing, 166

signal-to-interference-plus-noise ratio

(SINR), 26, 52, 58, 60, 61

signal-to-noise ratio (SNR), 4, 37, 58, 60

SIMO, see single-input multiple-output

capacity

single-carrier frequency-division multiple-

access (SC-FDMA), 288

single-input multiple-output (SIMO)

capacity, 7

single-input single-output (SISO) capacity,

6

single-user MIMO, see MIMO channel

singular-value decomposition (SVD), 46,

68, 86

SISO, see single-input single-output

capacity

sounding reference signal (SRS), 293

space-division multiple-access (SDMA), 13,

16

space-frequency block coding (SFBC), 291

space-time block coding, 69–70, 287, 291,

295

space-time coding, 68–70

space-time spreading (STS), 286

space-time transmit diversity (STTD), 286

spatial channel model (SCM) sector

antenna response, 168

spatial degrees of freedom, see spatial

multiplexing gain

spatial multiplexing, 9, 26

vertical BLAST (V-BLAST), 76

diagonal BLAST (D-BLAST), 66–67

gain, 11, 18

multiple users, 13, 79, 283, 292, 293, 295

per-antenna rate control (PARC), 63

single-user, closed-loop, 287, 289, 293,

295

single-user, open-loop, 283, 291, 295

vertical BLAST (V-BLAST), 63–65

spectral efficiency, see capacity

STS, see space-time spreading

STTD, see space-time transmit diversity

sum-rate capacity, 12

system performance criteria, 188–190

equal-rate, 229

equal-rate criteria, 189

maximum sum rate, 188

proportional-fair, 188, 190, 229

314 Index

system performance metric, 195

cell-edge user rate, 195

mean throughput per cell site, 195

peak user rate, 195

system simulation methodology

equal-rate, 202–213

proportional-fair, 194–201

terminal, 1, 163, 229

time-division duplexed (TDD) system, 162,

222, 229, 276, 282, 295

time-division multiple-access (TDMA), 22

rate region, 13, 15, 83, 125, 133

transmit adaptive array (TxAA), 287

transmit diversity

closed-loop, 287, 290, 292

open-loop, 69, 286, 287, 290, 291, 295

space-time coding, 68–70, 287, 291, 295

UMTS, see Universal Mobile Telecommu-

nications Systems

uniform linear array (ULA), 214, 216

Universal Mobile Telecommunications

Systems (UMTS), 285–288

uplink cellular channel model, 164

user selection, 130

vertical BLAST (V-BLAST), 63–65, 76

vertically-polarized (V-pol) antenna

elements, 213

waterfilling, 10, 68


iterative, 190–194

Worldwide Interoperability for Microwave

Access (WiMAX), 286

wraparound of cells, 174–176, 228

Wyner cellular network model, 159

zero-forcing (ZF) precoding, 129–141, 145,

194, 292, 296

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